.... that there exist things that unknowable, then would that proof constitute proof of the nonexistence of an omniscient god? If an omniscient being knows it, then it seems to follow that whatever "it" is cannot be unknowable, and if "it" is truly not known, then the alleged being is not omniscient.
Knowability probably needs to be defined in terms of the characteristics of the know-er. Presumably humans can know things that our smaller-brained ancestors couldn't know, and presumably also some creature with a larger brain might be able to know something that wouldn't "fit" into the human brain.
Similarly, their might be things that are just too complex for us to wrap our brains around, whereas some superintelligent aliens might be able to.
As for omniscience, it would presumably take unbounded storage capacity, plus omnipresence and omnichronicity to be able to acquire knowledge that otherwise would not be reachable or storable. OTOH, if there are only a finite number and size of things to know, and if is possible for some being to know stuff without having to acquire the knowledge, then such a being with finite storage capacity might suffice.