We just got a personal loan from the Commonwealth for our car. It is an unsecured loan, so I'm suprised they allowed it TBH, but they did.
The only catch was that DH had to be at his job for 3 months. He had 2 jobs at the time, the casual one he'd been at for 6 months, but they wanted him at the other one minimum of 3 months, as it was the higher paying job. That loan was only in DH's name, but my income from c'link was taken into account as household income.

Maybe you could have a chat to a different bank/credit union & see what happens? We couldn't get finance either, coz they wanted him at the job 12 months, rather than 3.