Sounds like a "modern" excuse.
1 - Paper logbook, and you sign a sworn affidavit with a notary
2 - Electronic logbook (I guess it shows the Hrs and what date they were entered, I am not so computer savvy, but would assume this can be verified )
3 - Your DFO/Chief Pilot/Fleet Manger would (assuming they were decent guys ) agree to verify your experience (approximately ) to any prospective employer, to give you a helping hand
I hope, as you would be happy to verify that they were fair even handed Bosses if you were asked to provide a "Reference"
I cannot believe that this should be problem, but, ask yourself . . . do you actually want to work for a company that is making it one , maybe better to move on & find someone who is more able to "get real".
I once had a similar problem, as my 2 most recent "former employers" didn't exist, so, having not had the foresight to request references on a regular basis, "just in case", it became pretty much impossible to get them, but, we found a solution. Maybe, they are not so interested in you & it is a convenient "excuse", another good reason to look elsewhere.