Even assuming that endocrine differences don't matter (same testosterone secretion, for example), probably not. The instinct of using appearance to determine group membership is probably too strong. I know black people who act very white, and they don't consider themselves at all white (why would they?) and neither do I. Black-white hybrids tend to self-identify as black even though presumably their brains are already half-white. I would expect one of these hypothetical people to think of himself as "a different kind of black man" rather than "a white man in a black man's body".

The biggest barrier is those black people are forced to be black for the rest of their lives. For example, they can act White, but they can't produce White or European offspring.

If you have a White brain but another race's body, you serve as the opposite example. You could behave black, but your children will still be White.

This doesn't really address the numbers problem, since it won't increase the number of white people.

I have scenarios for this:

1. White people who could have contributed to society but are held back by disabilities, can be given new bodies to be productive again.

2. Or the complete inverse. Non-whites who wish to elevate themselves or be apart of White civilization can get elective surgery that makes them White on the inside.

3. White organs can be grown in a lab and both groups in examples 1 and 2 can accept donations to re-enter society as a complete person again.