Я могла бы остаться здесь навсегда.
Imperfective verbs can't go together with the adverb навсегда. They do with the adverb всегда, because the 1st denotes a momentary manner of action and the 2nd a continuous/incomplete one. So one can say
Я могла бы остаВАться здесь ()всегда.
I believe it's useful to think of навсегда as
from this point onwards and of всегда as
always. And since навсегда implies a starting point, the verb it's associated with must be perfective, because these denote an action which has either started or ended, where a putative fixed point in time is evident.
Concerning politeness, which is true, in this dialogue останься could possibly work as well because the interlocutors seem to be close acquaintances, which allows for more informal address. Perfective imperative also sounds somewhat more persistent and demanding and in informal setting the degree of imperative is up to the speaker.