The atheist's dilemma as proposed by the OP was not a logical argument; nothing further to be said on that score. The basic question is sound though. At least as a starting point to jump off from. The dilemma is, in truth, one that everyone must face. Not just one subgroup.
The burden of proof, in the end, is really on yourself. To demand these proofs of others, either for or against God(s) is a dodge. It is an attempt to justify one's own beliefs either by accepting or discounting someone else's beliefs.
Or, worse, to demand an external proof is a reason to ignore the genuine question. I.E. to avoid doing the 'work' yourself. Whether it is religion, spirituality, or atheist, the only journey that can have any real meaning is the journey within yourself. It is easy to get bogged down in all the trappings, arguments, counter arguments, and so on, and so on. I fall into this trap myself too often, since I really do know better.
My suggestion. Stop demanding answers from someone else. Find the answers for you from within you. I'm not saying we can't learn from others, be it people, writings, mythologies. Absolutely dive into all of it as much as you can. Just remember the answer is not in all of those external things. The answer is in you.