Overkill in my opinion. SHA256 uses 64 iterations of functions over an 8 column array. SHA512 with 80 iterations over a 5 column array. There is no need to double hash because no one has been able to brute force either of these yet. Rainbow tables are not going to be useful to an attacker as long as there is salt.
Assuming 32 byte input (which is reasonable for your case - 20 bytes salt + 12 bytes password) my machine takes ~0,22s (~2^-2s) for 65536 (=2^16) computations. So 2^256 computations would be done in 2^240 * 2^16 computations which would take
2^240 * 2^-2 = 2^238 ~ 10^72s ~ 3,17 * 10^64 years
How long to brute force a salted SHA-512 hash? (salt provided)
So to really protect yourself, all you have to do is make the total size of your string large, like 32 bytes, and you wont even have to worry about that.
Check this out to understand just how crazy SHA-series functions are: