Using cash to upgrade SWISS Air economy (class X) flight booked with miles

Wondering if anyone has had success using cash to upgrade a SWISS air flight they booked using united miles? I've booked a round-trip from JFK <> GVA for myself and my pregnant wife using miles, and while I was able to snag business on the return the only option going there was economy. Seats together so we're happy to fly economy if that's the best option, but since it's a redeye and my wife is pregnant I'm trying to see if I can surprise her with an upgrade.

I have some cash in my united TravelBank so was thinking I could use that, and I can see on Swiss' website that there are business class seats available on the flight. But United says the flight isn't eligible for an upgrade.

Anyone have any success calling/chatting with United to try and book an upgrade? And if so, is this going to cost a fortune/are there other risks? Any advice is welcome!