You can give a bunny a butt bath with luke-warm water. You kind of sit the bunny in a small amount of water, you don’t have to trance it, just make sure he can’t kick out, that’s extremely important or he could break his back.
Then you gently rinse out the dirt helping with your fingers, if water alone is not enough you can use a mild, preferably organic, soap.
Afterwards you must dry him carefully but gently, a microfiber cloth is what I use. You can blow dry the butt on low heat, taking great care not to burn the skin.
Obviously it helps a lot if you have another person to help you hold the bunny.
If the skin is irritated and/or red you can use a bit of bag balm or other barrier cream on it.
You’re right in that you should never bathe a bunny, but a butt-bath is ok.
If he’s dirty due to sticky poop, make sure he gets enough hay in his diet. Or check so he eats the hay he gets, hay is key to bunny health

It is best to avoid wetting a rabbit, but sometimes that is the only way if they have a very dirty bottom. I don’t know what you used for the dry bath already, but here are instructions for a dry and wet bath. For the wet bath, you can either use a sink or if you feel more comfortable keeping your rabbit on the ground, you can use a plastic dish tub with a couple of inches of water. I haven’t had to do this in a while, but for the soap, I use what I use on myself – Clearly Natural bar soap. I cut off a little piece before I use a new bar. It’s hypoallergenic and not tested on animals. It comes in unscented or a few light scents.

http://www.bio.miami.edu/hare/buttbath.html

Well, bunnies sometimes do get poopy butt, and it’s not a huge deal if it’s just once in a while. Oxbow pellets and timothy hay is of course excellent food. Sometimes you need to adjust the amounts though, if the poopy butt recurs often, it’s likely you should cut back a little on the pellets and up the hay.

I have a bunny who I’ve had to give butt baths many times in the past. He doesn’t seem to resent me for it.