You'd need to redesign the engine housings and the rear of the airframe at the very least - the F-135 is 43cm longer and 3-11cm smaller in diameter, as well as 100kg lighter. The turbine system is also different, 1-stage high pressure and 2-stage low on the F135 and 1 & 1 on the F-119. With the different shape requires different inlet dimensions and shaping to get subsonic airflow, plus it's very likely that the various fuel, coolant, sensor, and whatnot other connections are all going to be completely different as well. That's just the engines...now, I imagine the avionics (being a digital glass cockpit) could be easier to swap out, though the only reason to do that is the EOTS system, which necessitates poking holes in the skin of the jet and wiring those up, and so on...
Honestly an F-22F could be done, but it really would be just like the difference between a Hornet and Super Hornet. A lot of parts commonality (landing gear, most of the cockpit, maybe most of the wings and all the tail/elevators offhand, but you'd have some definite changes to the airframe - and the F-22's stealth requirement means balancing that out a lot more carefully than the Super Bug does. It's not doable and it's frankly not a bad idea - good god, how many other aircraft have been rebuilt for modernization like that? Looking at you, B-52... It's just gonna be complex, especially because of the shape requirements for stealth capability.
Now, if you're gonna go that far, well, there's that Northrop/Grumman prototype still out there...anyone? Anyone? Bueller?