This morning for some reason I started wondering whether behave is related to have in the same way that become is related to come. After some research, the answer is yes, no, maybe, no.
Become is literally ‘come to be’: I came to be an ESL teacher in 2006.
Behave is not literally ‘have to be’: I have to be good/bad. Rather, it is reflexive: I have myself ?good/?bad; that is, I bear or comport myself *good/*bad/well/badly. There are two clues that behave is now a different word than be + have, if it ever was ‘the same word’. The first is pronunciation. The second is grammar: have is irregular – have had had, while behave is regular – behave behaved: *I behad well yesterday.
The prefix be– used to be more common and productive than it is now. A few months ago the Irish editor/language writer/blogger Stan Carey found himself Bewondered by obsolete be- words.