I claim that (again, in ZFC) P(C) iff P(D). Sketch of proof: 1. Up to canonical isomorphism there is only one complete ordered field. 2. C and D are complete ordered fields. 3. Therefore there is an isomorphism between C and D; in fact we can even write it down. 4. We can use this to build an isomorphism between C's complex numbers and D's complex numbers, and then between C's L-functions and D's L-functions, and C's elliptic curves and D's elliptic curves, and so on for every object required to state the BSD conjecture. 5. If we have a specific elliptic curve over D, these isomorphisms yield its equivalent over C (and vice versa); they pair up the groups of integer points in the two cases, showing that they have the same rank; they pair up the corresponding L-functions, showing that they have the same order of zero at s=1. 6. And we're done.

I claim that (again, in ZFC) P(C) iff P(D). Sketch of proof: 1. Up to canonical isomorphism there is only one complete ordered field. 2. C and D are complete ordered fields. 3. Therefore there is an isomorphism between C and D; in fact we can even write it down. 4. We can use this to build an isomorphism between C's complex numbers and D's complex numbers, and then between C's L-functions and D's L-functions, and C's elliptic curves and D's elliptic curves, and so on for every object required to state the BSD conjecture. 5. If we have a specific elliptic curve over D, these isomorphisms yield its equivalent over C (and vice versa); they pair up the groups of rational points in the two cases, showing that they have the same rank; they pair up the corresponding L-functions, showing that they have the same order of zero at s=1. 6. And we're done.

EDITED to add:

Although I stand by everything above, I can't escape the feeling that Kevin already knows all that and I'm therefore not answering quite the question he's meaning to ask. Let me put some words into Kevin's mouth:

Yes, yes, of course. Every mathematician who thinks about this stuff at all has something like that mental picture. But what really justifies that breezy confidence that that big pile of isomorphisms is really there? I understand that it feels obvious that none of the machinery you need to state something like the BSD conjecture depends on "implementation details". But this is the sort of thing mathematicians are good at getting wrong. It wasn't until the 20th century that we noticed how many extra axioms you really need to add to Euclid's system to make the proofs in the Elements rigorous. The axiom of comprehension probably seemed obviously innocuous until Bertrand Russell asked whether the set of non-self-membered sets shaves itself. A more isomorphism-y example: it seems transparently obvious that a set $X$ is the same size as $\{\{x\}\,:\,x\in X\}$, but this fails if you work in NF instead of ZFC. Maybe there's some implementation detail no one ever noticed we were assuming. How can we be sure?

Again, personally I'm very confident that I'd have noticed if some implementation detail were being slipped into anything in "normal" mathematics (or at least, I'm as confident as that I'd have noticed any other sort of gap in the proofs -- I don't think there's anything special here), and very confident that if I missed one some of the many many other mathematicians, some of them much smarter than I am, who have read the same textbooks and been to the same lectures would have noticed. But I think Kevin's asking whether there's some simple principle that makes it obvious without any need either to trust that sort of thing, or to check in detail through everything in the textbooks, and I want to be clear that this answer doesn't purport to give one; my feeling is that there couldn't possibly be one, any more than there could be some simple principle that makes it obvious (with the same restrictions) that there are no other logical holes in those same textbooks, and for essentially the same reason.