According to the Pri Megadim, one must sell his clothing for a Sabbath candle as well. (So the question doesn’t get off the ground.) His words are brought in Biur Halakhah, section 263.

You nicely proved that you are neither an ignoramus nor a fool. You can go back to your original pen name, just Tzachi.

עני שאין לו כסף לנר חנוכה ולנר שבת

Thank you, Binds and Sets Aside.
Honestly, I didn’t come to prove anything about myself. But I’ll seriously consider your suggestion.

Some time ago I received a similar question from my former student, Rabbi Bezalel Daniel:
Blessings,
I was asked whether one must sell his clothing for a Sabbath candle. The answer is that this is not explicit in the halakhic decisors.
By contrast, regarding the Hanukkah candle it is explicit.
So the Hanukkah candle is greater than the Sabbath candle.
But if he has only one candle, he should light the Sabbath candle, because of domestic peace.
Which implies that the Sabbath candle is greater than the Hanukkah candle.
A paradox.
One solution is to say that it was never actually stated that there is no need to sell one’s clothing for a Sabbath candle.
Possible, though it seems to me a bit forced.
I tried to think that perhaps there are different planes here:
publicizing the miracle is more important than my personal comfort, and therefore one sells his clothing. On the other hand, domestic peace is more important than publicizing the miracle. And so there is no a fortiori inference, because these are different planes. (One would need to assume that my personal comfort is more important than domestic peace. If I didn’t fear saying it, I would say that if I’m uncomfortable, there won’t be domestic peace, and therefore it would not make sense to sell the clothing for a Sabbath candle.)
I think this is nice, but in practice (something we don’t like talking about) it doesn’t help. Because I will sell the clothing in order to buy a Hanukkah candle. And then when I buy the candle, suddenly I will have one candle, which I will be forced to light as a Sabbath candle. So I will in any case be forced to sell another garment. And if I have no other garment, I will end up with a Sabbath candle and no Hanukkah candle.
I’d be glad to know whether the Rabbi has a solution for this.

And this is what I answered him:

My answer:
Hello Bezalel, and happy Hanukkah.
Mathematicians define transitivity as follows: if A<B and also B<C then A<C.
But not every property satisfies transitivity. For example, if A is B’s father and B is C’s father, it does not follow that A is C’s father.
Notice that in the definition of transitivity we referred to the relation “<.” There are other relations that are not transitive (like being someone’s father).
But even regarding the relation “greater than,” there is not necessarily transitivity, if one can say that A is greater than B with respect to criterion P (but not criterion Q), while B is greater than C with respect to criterion Q (but not P). Something like what you wrote.
An example of this: the relation of similarity seems, at first glance, to be transitive—if A is similar to B and B is similar to C, then A is similar to C. But regarding the labor of building on the Sabbath, Rabbi Avraham Borenstein wrote (in an article), and he is cited by Kehillot Yaakov (in the section on building and demolishing with utensils), that the primary category of the labor of building is building a house. There we gather parts together (bricks, wood) and create from them a functional space. This primary category has two derivative categories (according to Maimonides): curdling cheese—a gathering of parts that does not create a space; and making a tent—creating a space not by gathering parts. Notice that both derivatives resemble the primary category (in different respects), but between the two themselves there is not the slightest resemblance. So even the relation “similar to” can refer to one respect and not another (like the relation “greater than”), and therefore need not be transitive.
That is probably the solution to the collection of loops raised by the commentators (Tosafot on the lost object of one’s rabbi and oneself versus the honor of one’s father; Chatam Sofer on a similar loop in Berakhot; the well-known dilemma of matzah made from the new grain; and so on). There is no transitivity there, and therefore there is no difficulty in the fact that instead of a hierarchy we get a circle. In practice, of course, this can create a practical dilemma: what, after all, should be done first—what should be preferred? That is a conflict (what to do), but not a contradiction (that is, claims that cannot be reconciled and that it is impossible to hold together).
So in general there is no necessity that transitivity exist everywhere.
What you present here is not a contradiction (precisely because of what you explained, that there is priority according to different criteria, and therefore there is no problem with transitivity failing) but at most a conflict. However, you argued that this conflict has no way out. Notice that even if that is true, there is still no principled problem. We simply do not know how to act in practice, but that is not a substantive/theoretical problem.
Now we have to discuss whether there really is no practical solution to this conflict. What you are describing is that on a Sabbath during Hanukkah one must sell clothing in order to buy a Sabbath candle (because one does so in order to buy a Hanukkah candle, and then gives precedence to the Sabbath candle). On a regular Sabbath, of course, this does not exist. It is like “since it serves as a wall for Sabbath, it serves as a wall for Sukkot” only on a Sabbath during Sukkot. Here too: since he would have to sell clothing in order to buy a Hanukkah candle, he must sell it in order to buy a Sabbath candle.
But I do not see this as a problem. Here are a few thoughts I have on the matter:
1. Indeed, on a Sabbath of Hanukkah one must sell clothing in order to buy a Hanukkah candle, and then we will use it for the Sabbath candle, and now we will sell another garment and buy a Hanukkah candle. That’s it. The circle does not continue beyond that.
2. One could say that even on a Sabbath of Hanukkah one sells clothing for the lighting of a Hanukkah candle, and lights the Hanukkah candle and not the Sabbath candle. The reason is that we were already exempted from the Sabbath candle because we have no money, and for that one need not sell clothing. Once we are exempted from the Sabbath candle, we buy a Hanukkah candle and light it.
3. One more comment. When you light the candle that you bought with the price of the garment, you do not have to decide whether it is a Hanukkah candle or a Sabbath candle. Leave it to the Holy One, blessed be He, to decide. True, you won’t be able to recite a blessing, but blessings are not indispensable. And perhaps one could recite both blessings and let the candle serve both purposes (although that would involve performing commandments in bundled fashion).
Just one more clarification: a positive commandment overrides a prohibition. But what happens when there is a positive commandment versus a positive commandment, or a prohibition versus a prohibition? These are examples that you would not call a paradox, but there is still a practical dilemma (a conflict). The same is true here.
(In parentheses: when there is a positive commandment versus a positive commandment, or a prohibition versus a prohibition, there is a solution: passive omission is preferable. But I only wanted to illustrate the difference between a contradiction/paradox and a conflict/dilemma.)

The truth is that this ruling does appear in the halakhic decisors.
It appears in the name of the Pri Megadim in Biur Halakhah, section 263, paragraph 2, s.v. “at the entrances.”

A. Regarding the practical solutions to the conflict: seemingly it is obvious that since they said one does not sell clothing in order to light a Sabbath candle, then one would also buy clothing instead of a Sabbath candle. Meaning, a naked person who found some money in the street on Friday evening would buy clothing with it and not a candle. It seems that you do not accept the assumption that money and clothing are equivalent, but rather distinguish between selling and refraining from buying. What is the reasoning?
[In the first solution you suggested that he sell a garment (in order to light a Hanukkah candle) and receive money, and then buy a Sabbath candle with it. But according to the above assumption, the paradox remains, because with the money he ought to buy clothing for himself; Hanukkah takes precedence over clothing, and Sabbath takes precedence over Hanukkah.
In the second solution you suggested that after selling his clothing he should light a Hanukkah candle, because the Sabbath candle obligation has already lapsed due to its lower importance and does not reawaken. It’s a nice formalistic idea, but according to the above assumption the paradox still remains in the case of a naked person who has money in hand: what should he buy with it—Hanukkah, Sabbath, clothing, or Hanukkah?]

[By the way, what is the law regarding an ordinary Sabbath candle—let’s say there is an obligation to sell his clothing—if his wife does not want him to sell his clothing? Would domestic peace then require him not to sell it, even though that itself was the enactment: to buy a Sabbath candle for the sake of domestic peace?]