How do Buddhists who believe in a non momentary conventional self
I doubt many Buddhists would believe in a "non momentary" conventional self. I imagine most studied Buddhists would believe there can be a "momentary" conventional self.
during life account for karma without enlightenment?
Karma depends on the delusion of self. There cannot be ordinary (black &/or white) karma without self-belief.
Specifically, what happens to "my" karma when I die, if I am not enlightened and my karma has not been exhausted?
Karma will continue to reap results when the view of "I", "my" & "self" continue to exist. In the Buddha's teaching, only the false view of "I" dies. For example, MN 140 says "death" ("marana") does not occur when "I" is not conceived.
I'm not asking how to think of cause and result if the lack of self
identity extends from moment to moment in this life, because then "my"
karma would face a similar obstacle in this life (there being no
personal identity from moment to moment means it cannot be "my" karma
Sorry but the above sounds merely intellectual. If strong vexatious thoughts of "I" & "mine" continue to arise, these are personal identity.
Clearly most historical Buddhists
Buddhism became extinct in India and nearly twice became extinct in Sri Lanka. Clearly most historical Buddhists had terrible understanding of Buddhism.
believed that karma did not just exist in the suffering I cause
others, but that what comes around goes around. So if my life does
not, automatically without enlightenment, generate a new one, why
doesn't "my" death cut off that cycle?
Sorry but SN 12.2 defines "death" as the death of "a being" ("satta"). SN 23.2 & SN 5.10 define a "satta" ("being") as merely "strong attachment", "a view", "a convention". Many many suttas say the Arahants do not "die" ("marana"). That is why a synonyn for Nibbana is The Deathless. Buddhists who believe in "rebirth" merely spin around & around in speculative views & questioning forever.