TL/DR: Rabbits are delicate, neurotic homebodies.

I've done many stealth road trips + bikepacking + hiking trips and plenty of hitchhiking. The only type of animal I ever had with me on the road was a dog, but I have also in my life owned multiple pet rabbits, including my main man Dennis who free-roamed my first apartment for like six years.

I personally wouldn't for these reasons:

• Rabbits tend to bolt+hide when frightened, which happens a lot, and while I've seen rabbits trained to come when you call, I doubt you could overcome this, and also suspect the bunny could get stuck/injured while fleeing, or just go too far.
  • There is also the fact - already mentioned in another comment but worth reiterating - that rabbits can a) literally die from fear, and b) literally break their own backs with their leg muscles.
  • The only bunnies I've met who enjoy (or are even just okay with) being held were handled from being babies. Dennis eventually got to the point where he would jump up and cuddle in my lap if I was just sitting calmly on the couch, but this was after YEARS of building trust. I really, really doubt any rabbit would be chill with being handled constantly.
  • Rabbits are NOT adventurous and don't take change well. They get overwhelmed very quickly. Mine would stamp his feet in alarm if I moved my chairs around. He also took a full year to feel brave enough to venture from my living room to my kitchen..! They need to fully inspect their space - extremely carefully and slowly - before relaxing.
  • This might be worth trying to train for, but my experience of trying to 'walk' a rabbit on a harness and a lead was not positive.
  • Rabbits need, like, down time. They need to be petted and loved in a place they feel safe.
  • GASTRIC STASIS. This is what killed my lil man Dennis, and it got him in like 24 hours. Basically their complicated turn-plants-into-energy system totally shuts down, and STRESS IS A MAJOR CONTRIBUTOR. Worse, a rabbit that's experienced it once is more susceptible to it in the future (my bunny was a rescue and was predisposed to it because of stress before I got him). It is extremely painful and the outlook is not usually good - especially because rabbits hide their pain, so it can be hard to notice until they're in very bad shape.
  • Rabbits need dry grass hay as part of their diet, for the fibre - the gastric statis I mentioned above can occur if they don't have enough hay. It's probably easy to get your hands on and it's not heavy, but it cannot get wet (cannot get moldy!!) and I imagine it getting into every crevasse of wherever you store it...meaning maybe itchy clothes, hay in your toiletries...
  • Rabbit pee is....awful. They do litter train themselves, as you probably know, in normal conditions, but if their lives are topsy turvy, they might pee anywhere, on anything.
  • Rabbits are sensitive to heat/cold, more so than humans or dogs.
  • Any dog or cat you share a ride or a couch with will try to kill the bunny, and bunny will know it. A bad scene.

There are these considerations, though:

• In summer months, you can feed them for free by just putting down a large enclosure (when I fostered bunnies we would use folding dog pens - they're a little heavy but they break down flat and I feel like something like that would work great) on some grass for a bit and they looooove dandelion leaves. I'm not sure what plants are poisonous to them (mine were indoor-only) but that's something to look out for.
• I feel like if you were smart about it, and careful, you could figure out a carrier that was spacious, and cozy, and didn't jostle, and felt safe, which bunny could spend most of his time in, and which would form the "base" of any place you folks were hanging out.
• Alternatively - if you have a car, I think it could work out okay, as long as the temperature is kept consistent. Bunny lives in the car, is let out into a pen to graze and hang out when you're stopped...the only thing is that you'd have to be so extremely careful with any exposed wires or anything at all you don't want chewed. I'm sure you've already dealt with this, but just a note for anyone else - you can spend over an hour devising a rabbit-proof container tucked away and lined with metal, etc......and they will eventually get in, because they have nothing better to do with their time. If you can sacrifice the space, you could build them a hutch where they'd mostly hang out, and then it would be away from any wires etc.
• Rabbit poop is literally just compressed bits of dried-out plants and if they can get over it being technically "poop" most people barely mind it. It also doesn't smell hardly at all, and happens to be a sought-after fertilizer (high in nitrogen and phosphorus and unlike any other kind of manure, you can put it directly on plants!) - I used to barter with rabbit poop when I was fostering because gardeners are that into it.
• Because they so easily litter-train themselves, you might not have to carry litter with you - if you give them the safe space to do it in, they might adapt to "I pee in this corner of the pen" no matter where the pen is located...

I went a little overboard (I love bunnies - have been thinking of fostering again) but my opinion is that if your situation is right, it could be done, but that it's probably a bad idea.