I wouldn't recommend using SHA-2-512 for added security. Choosing SHA-3-256 would make more sense, as more knowledge has gone into the construction of SHA-3 than SHA-2, and the double bit length doesn't buy you anything if the fundamental construction is broken.
The primary design purpose of SHA-3 was not to replace SHA-2 with something 'better' it was to provide an alternate approach for hashing such that if one approach to hashing is compromised there is a completely different approach to hashing that is still viable.
There are good reasons to use SHA-3, but one should not consider using SHA-3 on the grounds that it is known to be a better hashing algorithm then SHA-2. It is not. In fact, SHA-2 has a much greater level of confidence due to it's age.
