I'm not sure this really constitutes an answer, but I think that it may not be fruitful to study the type of generalized question you discussed at the start of your post; you should be able to prove rather easily that for any real number $r>1$ you can find a sequence of primes $p_i$ with exponents $e_i=p_i$ such that the infinite product $\prod_{i=1}^\infty (p_i)^{1/e_i}$ converges to exactly $r$. Just pick the primes in a greedy fashion. Or maybe you use every prime, and choose the exponents in a greedy fashion. Or mix and match, and end up with many ways to represent your real in this fashion.
Given that there is so much choice and so little effect, it's my opinion that you're going to have to rely on the specific structure of your product to go anywhere fast, and perhaps this isn't so unreasonable of an opinion given the nature of almost all results we have about transcendence (namely that they exploit the very special definitions, structures, relations, etc., that their subject numbers enjoy, like $e$ and $\pi$). But hey, I'm no expert.