Cowboys of the Wastelands has a nice ring to it (don't know if taken). I picture a guy with a cowboy hat and a wastelander jacket riding a lowrider through a Mad Max wasteland.
Races
Not bad ideas, but no random mutations race? Also Elevated Lizards might go better with a wasteland desert theme (particularly if this is set in the Americas, where there are no apes), but apes could could work too if they're some sort of escaped pre-cataclysm genetic experiments that managed to thrive in the wild.
Classes
Lacks warrior classes. If Raider = Tomb Raiders (as in exploring pre-cataclysm ruins for goods) a term like Scavenger might work best. "Raider" tends to be used for ruthless wasteland warriors in post-apocalyptic fiction.
A "wastelander" class might be good just to throw that term out there. Might work as another term for "hunter" or "wasteland ranger/survival specialist". Or might be a more general warrior class.
Random mutations, as a PC? could be, since I guess there's still mutants being born, yes, I like it.
What's the biggest lizard in the US? I'm not going for a substitute for halflings. I figure there's enough chimps in your Zoos as to have them mutated by whatever, escape and then multiply over 200 years.
I figured the Drifter would be the warrior, kinda western but it's not gunslinger.
Scavenger... Yes, I like it.
Wastelander... Could be, instead of Drifter, and it's the warrior class.
Hunter because I am trying to avoid the Brave because it's not really a NA tribesman.
Ah, I figured that Drifters would be more like scoundrels or rogue types. They could work as warriors too, I suppose, but more like warrior-rogues, since the term has outlaw connotations. Or you could just change the name to Wastelander, like you mentioned, and phase them out, or use the word "drifter" more as a descriptive term rather than a class.
Having gunslingers would be nice too, if you're going for a western feel, but maybe you wanna keep the classes tight instead of going overboard. "Gun-slinging" could just be a skill.
Also, if you decide to just stick to the four races as character races, you might wanna consider the lizard men or other mutated creatures idea for antagonists. The wasteland is not just gonna mutate humans alone, after all. You could have giant crocodiles at the toxic swamplands, overgrown bearded lizards that shoot venom from dozens of yards away, psionic coyotes that can confuse their victims with their howls and cause them to hallucinate, etc. There could be a lotta stuff you could work with if you're going for a mutating wasteland that can warp creatures into newer breeds.