Just a thought, but if they destroy an entire ecosystem with Rotenone to eliminate an invasive species, then restock with new fish, the new fish will not be the same genetic strain that were there to start with, so can that really be considered a 'Native' species, or just another invasive strain that happens to be the same species? And what about the rest of the ecosystem? It will certainly be different when it comes back, so can that really be considered 'Native'? Poisoning a body of water to try to eliminate one species (which is usually unsuccessful, anyway) is similar to cutting your hand off at the wrist to get rid of a hangnail.
It's been my experience that nature usually does a better job of managing things than humans do, given half a chance. After a short adjustment period, most invasive species and Native species find a balance point to where they can co-exist. Nutria, carp, fresh water striped bass, brown trout, and other species have all managed to find their nitch. And, many 'Native' species could be also considered invasive, when they have been re-stocked far beyond their natural range, such as crappie, LG bass, freshwater striped bass, rainbow trout...
While I absolutely do not support introducing any species into a new area artificially, once it's done, it's done. There is no going back. The best you can do is just let nature work it out, and limit the damage on it's own. When humans interfere, we usually just make things worse. My advise is to just catch all the Rio Grande Perch you want and enjoy them. The problem will eventually be self-correcting.
Just my opinion, anyway.