To get the right to work outside a limited time instructing you'll need a "green card."
He won't be able to work at all in any capacity - limited or otherwise. He holds a CPL and therefore not eligible for a degree course with the option of OPT. However no mention is made of trying that route.
Even getting a flight school to sponsor an individual is pretty hard going. Sponsoring costs the company money, time and effort. Flight Schools run on a pretty tight margin and just can't be bothered with so many 'local' options available to them - especially when they know the individual is using it as a stepping stone and will be off in a couple of years.
Without the residency rights to work in the US, it is pretty tough with plenty of readily available competition.
Employers can simply get already qualified staff without the hassle of going down the visa sponsorship route enroute to a Green Card.
It is pretty tough out there in all walks of life not just aviation.
Not what you want to hear but that is the reality.