Both of mine are barkers... Oliver mostly inside the house, and Mya out. Mya's favorite thing to do is bark at the neighbors, and their dogs. It's a territorial bark, she has that "I own the world, and I didn't tell you you could be in it!" frame of mind. Fun. I can tell the neighbors on the one side are annoyed. She gets brought in the minute she starts barking (and I can catch her, lol) and often times if I know they are outside I will just take her out on a leash to avoid the hassle altogether. She is sloooooowly learning that barking means she goes inside - she loves it outside - so she's starting to get the hint.
I share this story because - I can relate. I can imagine being faced with such a decision. I'd like to say I'd never debark, but if it was a decision between letting one or both of them go and debarking, well I just don't know what I'd do. Oliver especially is so attached and has already been through so much (puppy mill rescue) that I don't know if he'd live through being separated from me and yet another rehome.
I haven't read the whole thread, but I did watch the video posted, and it broke my heart - gagging while eating or drinking or exercising for life? Yikes. I know it doesn't always happen and it's just one of those "surgical risks", but it's still something to keep in mind while making a decision.
I truly hope you're able to get a house you can rent so you both can rest a little easier. I have a lot of student debt and my living situation is not what I want it to be and I have no power to change it until some of my debt is paid down, so I can relate in that sometimes moving isn't always an option. Hopefully something opens up and it becomes one for you