Existence is frequently defined in terms of identity. 'exists(a)' ≝ '∃x(a=x)'

To assume that Existence is Identity is to assume that all things have concrete properties, which exist and can therefore be discovered. This is demonstrably false; at the fundamental level of reality, there is Uncertainty.

Only if you're an Objective Collapse theorist of some stripe. If you accept anything in the vicinity of Many Worlds or Hidden Variables, then nature is not ultimately so anthropocentric; all of its properties are determinate, though those properties may not be exactly what you expect from everyday life.

Quantum-level effects inherent in existence preclude the possibility of absolute knowledge of all things; there are parts of reality which are actually unknowable.

If "there are" such parts, then they exist. The mistake here is not to associate existence with identity, but to associate existence or identity with discoverability; lots of things are real and out there and objective but are physically impossible for us to interact with. You're succumbing to a bit of Rand's wordplay: She leaps back and forth between the words 'identity' and 'identification', as though these were closely related concepts. That's what allows her to associate existence with consciousness -- through mere wordplay.

Accordingly, I find that a better axiom would be "Existence is Probability".

But that axiom isn't true. I like my axioms to be true. Probability is in the head, unlike existent things like teacups and cacti.