Legion is a tribe that is bent on conquest, unending conquest. Caesar wants to use a city like Vegas to be his Rome, to make a nomadic tribe that swallows everything it consumes into a civilization that swallows everything it consumes. Caesar is a narcissist, he'll never stop and say OK, that's enough after he hs control of evey inch of the NCR.
But folllowing lore as is in the Fallout universe, the entire Legion wouldn't be able to handle the radiation and Super Mutants+Overlords simply because numbers can't do it and neither can a law that forbids Radaway or even Rad-X.
Like I've said before Caeser does not want DC AT THE MOMENT. No, he'll take the west coast first and then move on. This would take years seeing as he'd have to wage a war against the NCR in order to do so. He may not even win but lets say he does.
So Caeser wins, years and years later and decides hey, the west coast seems nice and goes on his mission to take it over. He finds DC. By this time the radiation levels have died down quite a bit ( radiation does go away, I do hope you know that), Mutants would have been hunted down by the BOS and killed. Leaving few in DC. Caesers biggest threat in DC wouldn't be the radiation or even the mutants, no, it would be the BOS.
Anyways, I'm gonna sleep, I'm tired. I'll get back to this tomorow morning possibly or after school/work.