It seems this doesn’t fully work out.

This is the language of the Talmud there:
“When I ask him something, he resolves it for me by reasoning; when I later find a mishnah, it refutes it. Rav Sheshet, when I ask him something, resolves it for me from a mishnah, so even if I do find another mishnah and it refutes it, it is a mishnah against a mishnah.
He said to him: Ask me something that I resolved for you like a mishnah. (Rashi: It was resolved for you by reasoning, and when you examine the mishnah, you will find that it supports my view.)
He asked him: If one cooked in part of a vessel, does the vessel require scouring and rinsing, or does it not require it? He said to him: It does not require it, etc.
He said to him (Rabbi Yitzhak bar Yehuda to Rami bar Hama): etc. It was taught: If one cooked in part of the vessel, the entire vessel requires scouring and rinsing, etc.
He said to him (Rami bar Hama to Rabbi Yitzhak bar Yehuda): If it was taught, then it was taught.
And what is the reason? The verse says (regarding a sin offering): ‘And if it was cooked in a copper vessel… it shall be scoured and rinsed in water’ — even if only in part of the vessel.

A. So it is clear that Rami bar Hama does pay attention to mishnayot, of course. Rather, Rami bar Hama assumed that his reasoning was consistent with tannaitic sources that he did not know. I described the word “matnita” as mishnah, but I was mistaken, and apparently there in the Talmud that term includes baraitot as well (“it was taught”), and Rami bar Hama also takes those into account and withdraws his reasoning because of them.
B. And the fact that Rami bar Hama was less expert in baraitot — not mishnayot, as I mistakenly wrote in the question — is indeed what it says there. But that was not Rabbi Yitzhak bar Yehuda’s direct claim. Rabbi Yitzhak bar Yehuda did not claim that Rami bar Hama answers by reasoning and therefore there is a greater chance of finding a contradictory tannaitic source, unlike Rav Sheshet who answers from a source and therefore there is less chance of finding a contradictory tannaitic source. Rather, Rabbi Yitzhak bar Yehuda claimed that in the event that one finds a contradictory tannaitic source (whether to Rami bar Hama’s reasoning or to Rav Sheshet’s other tannaitic source), from his perspective there is a difference. That implies that Rabbi Yitzhak bar Yehuda’s claim stands even if the chance of finding a contradictory source is identical both when Rav Sheshet answers and when Rami bar Hama answers.
C. You suggest an interpretation according to which, if Rav Sheshet answers from a tannaitic source and then Rabbi Yitzhak bar Yehuda finds a contradictory tannaitic source, one harmonizes them in a way that actually makes use of both sources. I don’t think that is the meaning; rather, the contradiction is absolute, there is a tannaitic dispute, and one must choose or determine the conclusion. Because if harmonization were possible, then Rami bar Hama’s reasoning could also be reconciled. Or do you mean that although reconciliation is possible, one does not do that for mere reasoning, but only for a contradictory baraita? If so, why? And in any case, in the example discussed there — whether cooking in part of a vessel requires scouring and rinsing — the contradiction seems absolute, and there is no room at all to reconcile it.
D. So I am left with your answer: “Even if there is no resolution to the contradiction, you still have two mishnayot that disagree, and your reasoning has something to rely on.” What does that mean? This is not about practical Jewish law, where the questioner is afraid of violating prohibitions and if he has a tanna to rely on he can calm himself that he was entitled to rely on that tanna even if in truth the other tanna is the one who is right. Because there the discussion is about cooking the most sacred offerings in a vessel (though perhaps one learns from this to other prohibitions?), and in Babylonia at that time there were no offerings. Nor does “I can maintain my claim” apply in matters of prohibition. Rather, this is simply about Rabbi Yitzhak bar Yehuda wanting to know the halakhic truth. And that is the question: if the chance of finding a contradiction to Rav Sheshet’s source is the same as the chance of finding a contradiction to Rami bar Hama’s source, then in the end asking Rav Sheshet rather than asking Rami bar Hama does not improve Rabbi Yitzhak bar Yehuda’s chances of knowing the halakhic truth.
I came back in order to focus the question on the last sentence of your answer.

This is not a matter of relying on it despite it being mistaken — which really applies only to practical Jewish law — but of relying on the reasoning as correct. Even if we are dealing with Jewish law not in practice, support from a baraita or mishnah still gives significance to my reasoning.

Do you mean that if it is one baraita against another, then each one has, say, a 50% chance of hitting the truth, whereas reasoning against a baraita is, say, 1%? So Rabbi Yitzhak bar Yehuda is saying that with Rav Sheshet he gets a “certain” 50% at least, whereas with Rami bar Hama he does not?

That is one possible model.
Another possibility is that if there is a tanna who agrees with you, then the reasoning is clearly correct; it is just that there may be other reasonings that also need to be taken into account.