is Billie going to stay when Chibnall takes over with a new Doctor?

I agree. Sure we haven't seen her yet and she might be fantastic, but I think we need a completely fresh start like Series 5 was. New Doctor, new companion, new TARDIS, new logo etc. New everything.

I might end up missing her but seeing as Peter is leaving the only leftover of the Moffat era will be Bill. I'd rather have no leftovers. I remember RTD saying when he was leaving that old companions were unlikely to return because it was "time for Moffat to come up with new characters". I hope the same will apply here.

Basically I'd rather Chibnall gets to start with an 'Eleventh Hour' rather than a 'Deep Breath'. The Doctor meeting a companion for the first time is such an integral part of the show, and an iconic moment. For that not to be a part of the start of Series 11 would be a wasted opportunity. I'm sure Chibnall is itching to write his first companion meets The Doctor scenes and would want to create and cast his own characters.

Personally the thing I hated about the RTD to SM transition was that there was nothing connecting them so it felt like an entirely new show which as I'd liked RTD era meant that I got annoyed of a lot of sudden change.
That being said I have a feeling I won't like Billie (or is it asBillie) there little taster clip of her made me almost hate her as I just found her annoying and grating. Now perhaps that's wrong perhaps she'll be wonderful but I've got to be honest I've said that about a lot of things during SM run and been constantly disappointed so let's see.

Not basing this on anything but my guess is she's probably going to leave. Same reason you said.