I don't see where dualism comes in. Specifically what kind of dualism are you talking about?

Being a monist means believing that if things don't make sense, it is due to limited knowledge and a limited brain. But the problem of existence is such that no amount of knowledge will solve it: there's nothing we could ever learn (or even believe) within X that would solve this problem. ... So to the extent that I am correct that this problem is not in theory solvable in X means that X is incomplete.

A problem being unsolvable within some system does not imply that there is some outer system where it can be solved. Take the Halting Problem, for example: there are programs such that we cannot prove whether or not they will never halt, and this itself is provable. Yet there is a right answer in any given instance — a program will halt or it won't — but we can never know in some cases.

That you say "I cannot understand what the answer to the problem could possibly be" suggests that it is a wrong question. Ask "Why do I think the universe exists?" instead of "Why does the universe exist?". I have my tentatively preferred answer to that, but maybe you will come up with something interesting.