Why? Every general that did rebel against the Emperor planned on conquering the entire Empire.
Not true of Carausius. He issued coins in his and the legitimate emperors' names -- he basically wanted tio be recognized as one of them, with Britain and a small part of Gaul as his share (Diocletian et al. weren't having any of it).
To fill in a few gaps:
Britain was a lot more trouble than it was worth for a long time -- it had been all or part of breakaway empires for most of the period from 260 - 411 A.D., when the split became final. [The rebel emperors were Postumus, Carausius; Allectus; Magnentius; Magnus Maximus; Marcus; Gratian and Constantine III. Wikipedia and Regnal Chronologies have their dates. Also Constantine the Great, who started in Britain and eventually succeeded in taking the whole empire]. It's not surprising the empire gave up on the place.
The heresy from Britain was Pelagianism.
We know of two appeals from Britain to Rome, one addressed to Honorius (d. 425), the other to 'Agitius' (probably means Aetius (d. 454) since the addressee is described as three times consul, but could be Aegidius, who ruled in northern Gaul in the 460s). In any case, Honorius wrote back to say Britain would have to defend itself, and nothing came of the second request.
It's true that there's not much in the way of contemporary narrative sources for Britain between 411 and the mid-500s. The Life of St. Germanus is the only one I can think of, together with a few mentions in continental writers like Prosper of Aquitaine. However, we know (A) how things stood in 411 (a Roman Province largely denuded of troops) and, more or less, (B) how things stood around 550 (a bunch of British kingdoms existed in the West and North, Picts were further north than that, and parts of the south and east were held by kingdoms identifying as Angle, Saxon or Jute). There are only so many plausible routes from A to B.