Whether or not it would work depends on how well the 12V-only PSU would cope with having single-rail load and how gracefully it might handle load-sharing.
1- worst case: the 12V-only PSU cannot operate heavy 12V load without any load on 3.3/5V rails, trips an over-voltage protection which activates crowbar circuitry on the 12V, shorts out the 12V rail, the heavy combined current ends up burning PCB traces on the GPU, PSU wires and possibly the PSUs themselves.
2- voltage regulation between PSUs gets messed up due to fluctuations on the 12V rail as both PSUs try to level off, the main PSU's 3.3/5V outputs end up out-of-bounds because of it and you end up with crashes, reboots, etc.
3- everything goes well and it actually works.
Since normal PSUs are not designed for load-sharing, the most likely outcome will be somewhere between #2 and #3.