I’m not trying to mount a proof of inspiration. I’m simply trying to say that a belief in the inspiration of Scripture is not incompatible with a recognition that Adam wasn’t historical.
Your initial question seems to presuppose a particular view of inspiration. Namely, that on every subject the Bible touches its treatment of the subject matter should be 100% accurate by all criteria (scientific, historical, theological, etc.). If that is the view of inspiration that one holds, then it makes perfect sense to say “If Paul is not reliable on X, inspiration is disproved. Therefore I cannot trust him on Y.”
Your question really boils down to “Why should we believe the Bible is inspired at all?” If you used to answer that question with “Because it is 100% accurate about everything it says as judged by any criteria” then you will probably need to find a new answer.
Chris Massey in a comment on an article at BioLogos.com about Paul’s view of the historicity of Adam.