BIOS mod to support PCIe 3.0

[anw]

I have an older computer running 3.2Ghz AMD 64x dual core, 4gb DDR2 ram, and a 240GB SSD. The model of the PC is a Shuttle XPC SK22G2 with motherboard FX22 v1.3. I've upgraded everything to the maximum except the GPU. I picked up a couple small form factor PCIe 3.0 cards, one nVidia GeForce 745 and one AMD R7 260. Neither can be detected on my PC. Official bios updates are out of the question. It is my understanding that PCIe versions require different power ratings and mine is either PCIe 1.0 or 1.1. My power supply is 300W which should be enough to run these low profile cards however.

I guess my first question, is it theoretically possible for someone with the right skills to modify the BIOS to accept these newer cards? I know the optimal solution is just to buy a new system blah blah, I understand, but I'd rather not do that at this time. 

I would be willing to commission someone privately to help me do this, I am quite knowledgeable and familiar about computers so I could follow instructions to help me do it myself as well. I just know how sensitive modding BIOS chips can be and the high risk of “bricking” my PC by doing so, therefore I seek guidance.