If I understand @xmunk@sh.itjust.works's comment elsewhere in this thread properly, I think that's what a pause interrogative may be. I also agree with them that it (and the interrogative start) does better fit some ways of speaking.
Yeah, that's either two "sentences" or one statement imo!
The first part doesn't even need to be a question. A suggestion like that would usually be a statement. If there's enough rising intonation that it needs a question mark, there's probably enough of a pause to justify having two sentences.
It's not about making it a question, is about showing doubt.
"Jake should've been there last night (?), but I doesn't have time to check."
Sure there are ways to phrase that differently, but it's the sort of message we can easily communicate with hand gestures and intonation, but fail with written word.