I'm a bit weirded out by the implication that a purely abstract number system could be possible or impossible based on the actions of any entity, even a god. That's the only way I can see for God to be glorified in the way described. Yet that would mean that it would be possible to create a universe that has free-thinking mathematicians but in which surreal numbers could not exist even as a concept. Which is quite a claim.
I don't think that's so weird. An omnipotent entity controls the rules of reality. It could construct a universe in which contradictory things are true, or whatever it wanted. This is a bit like the puddle assuming the world was made for it because it fits so well into it - we take the rules of our reality for granted but that need not be the case in this thought experiment.
The rules of reality apply to reality, though. It's easy for an omnipotent power to make physics do seemingly contradictory things. But pure logic is not part of reality.
You could talk about how a real-life implementation of some element of math doesn't match the logical version derived from a specific set of axioms, but there's no real-life version of surreal numbers in our universe anyway so that's basically true already.
Are you so sure that logic is separate from reality? A great deal of logic depends on physics. Reversing the arrow of entropy, for example, pretty much breaks 'if x, therefore y' statements.
