Presuppositionalism is the last bastion of the sophist.
Yes, you can no more use logic to prove logic than you can use perception to disprove solipsism.
However, the question is irrelevant.
Until and unless we encounter a qualia which contradicts that fundamental assumption of reality (i.e., it exists) we basically have no choice but to assume it exists.
I can no more ignore logic and perception than I can ignore the fact that I exist. Even if I am a story being told by an inchoate unknowable devil god for its own amusement and not an actual being actually experiencing events and part of a reality, the fact is that my current state is indistinguishable from my existence being true.
Once I accept that my existence is true (and the only other option is not an option at all) then everything else (perception of the exterior universe, the logical absolutes, mathematics, science and so on) follows irrevocably.
Stop at any part of the chain and say “well, what if everything we know is wrong?”
To which the reply is: “So what if we’re wrong?” Does it really make a difference?
People are wrong all the time. We have, in fact, as a species, been unbelievably wrong about basically everything. From heliocentrism to Newtonian mechanics to relativity, all of our “known” facts and truths have proven to be wrong by latter generations. Presuming that we are somehow the generation that got it right is the height of arrogance. There is no doubt in my mind that quantum theory will, one day, be viewed in the same way the Newtonian gravity is today.
But regardless of whether or not we are wrong in the absolute sense, wrong is not a binary state. It is possible to be less wrong. A person who says the Earth is a sphere and a person that says that the Earth is a flat disk are both wrong; but one is much less wrong than the other.
Similarly, when it comes to logic, we can be fairly certain that we are correct to assume that A == A, A =! !A and so on are much less wrong than other options. Sure, we can’t “prove” that. All experience until now points to those two statements being true, and if one day they prove not to be true… well, then they’ll not be true, and we’ll go from there.
But asking for absolute proof is ridiculous. Nobody has absolute proof of anything and nobody can. Anyone that claims the opposite is selling you something.