lol Ive already done something like this to my psp. It didn't work well, but it did extend my range by a few feet. the antenna I added was made of aluminum foil and placed under the battery compartment. The thing is you cant get many signals out of the psp like you can out of a phone. PSPs don't support the 4g network. they use similar protocols, you had that part right, but there are some noticeable differences. Sprint doesn't want anyone to just use their network with a device that they didn't approve of. When A company makes a phone and plans to put it on the network, there is legal papers to sign. Sony and sprint don't work together on this part well because Sony would have to pay a fee and put sprint on the PSP. Now wouldn't that be ugly with a PSP door printed with "PSP powered by Sprint"? I think this was part of the reason sony doesn't make PSP phones. consider this idea dead unless you plan on putting a router and modem in your PSP. Another approach would be putting one of the new phones into your PSP. You know, the type that works as a hotspot. I think they are from Verizon. Anyway you can search up these phones. Im guessing that they're around $100 to $300 USD and have a touch screen.
BTW welcome to Acidmods