Welcome to the forum.

We can speak of necessity accruing either to facts or to descriptions/sentences/propositions. The latter are representations of the former; facts are something concrete.

When we speak of 'truths,' it's ambiguous between these two categories. Sentences, for instance, can be true or false, and when they are true, it makes sense to speak of the facts to which they refer as truths.

I think you're right that necessary truths in the sense of descriptions/sentences do not explain anything, but it may still be that the necessary facts which correspond to them do explain things. For instance, if God exists necessarily, then the fact that he exists is necessary and explains things. So I don't think it will be enough to rule out that necessary truths in the sense of necessary descriptions/sentences are explanatory (because your interlocutor's argument would be best if he meant something else) and I don't think it is possible to rule out that necessary truths in the sense of necessary facts are explanatory (because you want to show that one of them is).

I am not sure what to say. I would revisit the argument you are presenting and see whether it implies the existence of a being with some of the properties God has traditionally been taken to have; if it does, then you should be able to find some reply to your interlocutor.

I'd also note that there 'explanation' seems to be ambiguous, and we can't expect a principle of sufficient reason to hold for every sense of the word. The fact that a circle cannot be squared can, I think explain some things, like Bob's failure to square the circle. And arguably some mathematical truths (first mathematical principles, if there are such things) explain others. And in, e.g., scientific inquiry we speak of 'inference to the best explanation'--as though there were more than one! I personally suspect that if there is a defensible version of PSR, it will be appealing to a distinct though of course related sense of 'explanation'.