Isn’t there also a problem here מצד deriving benefit from another person’s Sabbath violation?

And maybe there is also something like the prohibition of telling a non-Jew to do something?

I don’t think so. The brother is doing it for himself, not for him. And regarding the prohibition of telling a non-Jew, simply speaking there is no prohibition when the instruction is given on a weekday (except according to Rashi, who holds that the prohibition of telling him is due to agency for a non-Jew, as a stringency. And with a Jew they were not stringent about this, since with a Jew there is no agency for a transgression. However, according to my own approach one could discuss whether, with a Jew who does not believe, there is agency for a transgression. It seems to me that simply speaking there is not, though this can be analyzed further).

The title really does not reflect the question.

You wrote: “If he does not believe in Jewish law at all (and not merely that he does not observe the commandments), then in my personal opinion there is no problem, because his transgressions are not transgressions.”
What do you mean? That since he does not believe at all, it makes no sense to talk to him about what is permitted and what is forbidden?

If we borrow an analogy from civil law, then for example: is it like saying we cannot tell a Neturei Karta person to obey the laws of the state when he does not agree with its very existence?
But at the end of the day, in civil law we would still punish such a person even if he does not recognize our authority.

In short, what is the Rabbi’s reasoning for his idea??

It has nothing to do with civil law. In civil law, commandments also do not require intention, nor do they require belief. The point of the law is not the service of the lawgiver, but preserving order and fulfilling obligations. In Jewish law, its point is the service of God, not a collection of actions done absentmindedly.
Regarding whether commandments require belief, see my article here:

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