did not test that but here is a page how to "google" the "de-googled" os for supporting that
https://doc.e.foundation/support-topics/android-auto

because of that lineage could be a better option

About Fairphone: there is an alternative (Shiftphone) that is more expensive but with the main plus points of having a higher storage option and the mainboard is replaceable, they also have somewhere a cheap (~200€) phone that should work if you really just need a phone
(i mention that as an option, because having choice is always better even if it ends up being the first thing that gets choosen)