I am not sure what you mean by continuously supportable. Saipan was the main base of the nuclear planes, as well as the conventional bombers.
The island hopping campaign was necessary for both. The early nuclear bombs were large, and could not be launched from a carrier based plane, or even the B-25 which was carrier based in 1942. (a one time deal, I think all of them were lost on that mission.)
I don't believe the US had any more nukes after the first two were deployed for at least another 6 months.
The B-25 was never carrier-based. There was one incident (with many modifications required) where a group of B-25's flew off the deck of the carrier Hornet. This is the famous 'Doolittle Raid'. It was a psyclological raid on Tokyo. Some of the planes flew (real low) over the imperial palace but, with orders not to cause damage to it. It was a one-way trip where the pilots were to ditch in friendly Chinese territory. Some made it, many didn't.