We have traveled with 3 cat, not all 3 at one time but did have a dog and 2 cats together.
The biggest issues are indeed being cooped up in tight quarters, the possibility of them getting out and the fact they are not friends. We found you will also need to put a guard on your entry door screen so the cat can't pull loose the screen and get out (been there done that with ours). Also your window screens..in our rig they slide too easily and the cat can easily pull them aside and get out if the window is open. We used painters tape to secure them but a more secure latch would be more ideal.
If I were in your situation I'd find a home for some of them and only keep the two that get along or may find a home for 2 and keep one.
To manage the worry about them getting free...it is indeed a worry and we had one escape but luckily once she hit the ground she found out she was not somewhere familiar and we could grab her.
We rigged up a small dog harness (more secure than a cat harness) and put a dog collar on them (not a cat quick release collar). The collar has to be tight enough it will not come off over the head. We hooked a leash to the collar ran the end through the loop of the harness and put in a knot, leaving enough distance it would not restrict neck movement. We left this set-up on while we were rolling but when parked we took off the leash. The leash gives you that last bit of something to grab in case you need to.
Not all cats will love being leash trained or like having the collar on, a cat can wriggle out of anything you put on them. Get the cat or cats used to the harness, collar, leash deal at home, allowing them to drag the leash around for a short while. If they get used to it, try taking them out for walks in the yard then expand to the driveway, etc. Some cats will walk ok if there isn't anything around to spook them. We often took our cat Emmy out on a leash potty walk at an RV campground they were familiar with. But the older cat we had would not stand for walking on the leash and even fought having the harness. Our older/1st RVing pets have long since passed on. The new cat hates the harness and will not walk on a leash.
good luck with whatever you are able to do.
To the poster that suggested taxidermist...how cruel to speak so rudely to the OP! If you can't stand having cats that is your choice but to even joke about something like that is in poor taste!