The underwriters of the new mortgage will want to know that OP isn't paying the old mortgage. If he typically pays the old mortgage, his debt to income ratio will skyrocket, probably beyond any amount that could possibly qualify for this mortgage. What they're asking for is proof that someone else is typically paying the old mortgage. Not to say that fact alone absolves OP from any responsibility regarding the mortgage on the house his ex-wife lives in: he could very easily find himself paying for both houses, or bankrupt because he can't afford either. It's quite a mess he's gotten himself in, which would be compounded by adding another mortgage and/or the ex-wife failing to pay the previous mortgage. That's why underwriters ask for things like this. A reasonable person would be willing to provide that proof to the OP, but apparently one or both parties is not reasonable.
If I were the lender, I wouldn't lend OP the money whether she was current on the payments or not. There is a risk that he might be legally obligated to pay his ex's mortgage as well as his new mortgage sometime in the future.
I agree with everything you said except for "it's quite a mess he's gotten himself in".
Let me give you a real life example that i've given here before. I know a couple who were amicably divorcing. Both spouses had good jobs and worked at the same place I do. The wife did not want any of the husbands retirement funds. When the judge (the same old fart I had) reviewed their settlement agreement, he rejected it and told both parties that the husbands retirement account must be divided because she' is entitled to it. The wife held fast that she did not want any of the husbands retirement account and after months of delay, the judge finally relented and granted the divorce.
If OP's judge were looking out for him the way the judge did the wife in my example, OP would not be here asking why he's getting screwed over.
Hopefully OP has not executed a quit claim deed to his ex-wife, but I'm certain OP's judge didn't overlook that detail.