I initially wrote this as a comment, but it got too long and it sort of contains an example, so here goes. Reflection seems false in a number of contexts, since there are many properties that can't be satisfied in any canonical way. For example, there isn't a small or simple basis for the reals over the rationals.
But maybe more in the spirit of the question than constructions that require the axiom of choice are a number of strange Banach-space counterexamples that are built using tools such as a sufficiently fast-growing sequence, a concave function that tends to infinity more slowly than any power, an injection from finite sets of rationals to the positive integers, etc., where the properties you need can be achieved reasonably simply, but not canonically, and the combination of the various elements is best viewed not as a single example but as a technique for building examples, where the precise details of the implementation clearly don't matter.
I'm making a slightly stronger point than may immediately be apparent, which is that for some of these strange Banach-space properties (a famous example being the property of not containing $c_0$ or any $\ell_p$ space, which was first shown to be possible by Tsirelson), not only is there considerable flexibility in how you build counterexamples, but it appears that this flexibility is in some sense "necessary". One way of making that assertion semi-precise is to say that there don't seem to be additional (sensible) properties you can insist on that cause the flexibility to go away.
I'm not saying that Reflection is definitely false for this kind of property, but it does seem to be, and I see no reason to suppose that it would be true.
