Well I haven't seen a desert rose come back like that, EVER???? I'm really not sure where you should cut out the rot??? The goofy thing should have been dead by the looks of it, but it has new healthy leaves and flowers????
I saw what you meant about it sort of seeping fluid, (which is poisonous, by the way) I wouldn't be touching that, unless you are going to make poison arrows
I "think' I would start at the top and just take the absolute deadest stuff off, then leave it and see how that goes. Do it slowly in stages, you might have to cut on an angle, by the looks of the dead parts.
If I kill your plant, YIKES!!! I will take all the blame, lol.
Seriously though just go real slow, on the pruning. There was a very extensive thread on the desert rose plants awhile back, if you look and try and find it, the person on there had fantasic desert roses. You might contact them, and send your pictures and information, BEFORE pruning??? Hopefully they can give you more help than me?
If you do hear from them, please let us know what they say? Would really be interesting, to know what they say. Good Luck.