The -ee ending seems in English to be passive in sense, since we have pairs like employer/employee; but the passive use derives (in French, Latin, and English) from the fact that it's a past participle: I have escaped. With a transitive verb, it serves both grammatical functions: I have employed; I have been employed.

I regard escapee as the plain English word for it, and wince at the ugly escaper occasionally substituted for it.

Escaper sounds like a neologism, because it's so unfamiliar, though in fact it's older, and escapee dates (in English) only from 1865.