If I were the spouse, I wouldn’t want the mortgage only in my name. The house is owned by both of you, so the debt should be shared. Is this other debt joint to you and your spouse? If that’s the case, perhaps there’s a case for an exception. But if your spouse is uncomfortable with having the debt in their name only, it sounds like you’d be trading one issue for another.
I think this situation really points to the fact that you need to clean up that debt as soon as possible. Skip the refi. Dial back your 401k to only get the employer match, cut you expenses, and point everything you have at the debt. Set up a debt snowball (either by interest rate or amount outstanding), and start knocking things out. It’s not a fun situation, but when you get out of this, you should be in a very strong position.
Also, the PMI will eventually drop off, so just keep an eye to your equity and drop it when you can.
You’ll get there. Sounds like you weren’t totally sold on the refi anyway, so this development seems to take you back to the original plan.