Hello,
Yes, any gcc country member has the right to extradite anyone accused of breaking their laws. I was too in your husband's situation, but in my case I never moved. I accumulated quite a debt (caused by my sudden termination from work and freezing pf my accounts due to late iqama renewal), and the bank threatened to hand my file to the Saudi credit bureau, which have the authority to have people arrested. My card had expired. I found work and negotiated with the bank, and they agreed to receive monthly installememts of my debt until it was done, and that's what I did.
I'm afraid to say there is no way out of this. Of your plan is to to stay in the GCC, I recommend you find a way to pay back the debt.
Good luck. 🍀