Everything sounded perfectly good until the last bullet:

why believing that things like a first uncountable ordinal can contain reality-fluid in the same way as the wavefunction

ERROR: CATEGORY. "Wavefunction" is not a mathematical term, it is a physical term. It's a name you give to a mathematical object when it is being used to model the physical world in a particular way, in the specific context of that modeling-task. The actual mathematical object being used as the wavefunction has a mathematical existence totally apart from its physical application, and that mathematical existence is of the exact same nature as that of the first uncountable ordinal; the (mathematical) wavefunction does not gain any "ontological bonus points" for its role in physics.

or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness

Pinning down a single model up to isomorphism might be a nice property for a set of axioms to have, but it is not "reality-conferring": there are two groups of order 4 up to isomorphism, while there is only one of order 3; yet that does not make "group of order 3" a "more real" mathematical object than "group of order 4".