Rotting bulb possible solution
 

I keep the bulk of my orchids in shade houses, I'm keener on them having as much as the natural environment can offer. The problem was during our rainy season I would have some bulb rot. (6 of one 1/2 dozen of the other-in the inside greenhouses the orchids sometime ssuffer from dehydration and that's not good either) I noticed my unpotted divisions that were temporarily housed in empty pots had few to no issues. I began to switch everyone over to that method. Some had to be repotted, they didn't take kindly to being unclothed at all(!) Most thrived in ways I never expected. I can move them all easily and it saves me a lot of repotting time. When I fertilize or give nutrients, the entire orchid benefits. I do keep the terrestrials in medium, in pots. Now, I am in Florida, in Sarasota, so consider your weather conditions, and if you want to try this method, do a few at a time, to make sure who does well and who doesn't. It does take time.

Great news!

I'm in central Florida and I do the same thing! Not with all of them but when I have one that needs repotting, I just yank it out of the pot and water it in an empty pot....mine have all done quite well. I've been repotting in lava rock lately too. That's made a huge difference.

Once a bulb starts rotting, is there any way to stop it without chopping up the plant?

I don't think so.
Physical removal of any rotting portion as early on as possible is the key in saving a rotting plant.