What I would do:
First, wash his bum. That doesn't look that bad, I think you'll get away with using a wet washcloth since it's a rather small area. Use plenty of water though but try to not get too much of the rabbit wet.
I don't use shampoo or soap, I don't mind some discolouring but prefer to avoid having to wash all of the shampoo out.
My old girl does need butt baths, I do that in a rather narrow container, with one low side (2-3"). At the low side I put stuff and a towel so that platform is level with the container wall. I put a rubber floor mat in there so there is some grip.
Water appr. 1" deep, 30-35°C, with a bucket of fresh warm water ready and na second to dump the water now and then.
Rabbit goes with it's rear legs and bum in the water, his front half resting outside. Works for me, I can put the left hand on the head, give treats, and work the soiled area with the right.
I need to change the water 2-3 times since my old girl gets a lot of poop stuck.
Then I place the rabbit on sheets and towels and try to gently rub as much water out of the fur as possible, do the last round with kitchen paper and then get her dry with a hairblower (thank god she doesn't mind the noise), always fingers in the airstream to not let it get too hot. To get between the legs and at the nether regions I put her on a piece of wire floor since she has athritis and I avoid getting her into uncomfortable positions more than necessary. This takes some time since she is all wet there, so I take breaks and make it as comfortable as possible (picture).
Well, and then I would cut his fur down there short. I use a pet trimmer, best investment ever. Not to the skin, but I wouldn't leave more than a fingers width where the fur was dirty. For me the easiest way to do that is putting a towel on my lap, placing the rabbit between between my legs on it's back (not sitting on the edge of a chair so it can't fall through), I need to wear a thick, old winter jacket since my girl is a biter when frustrated and goes to work at my arms while I'm giving her the haircut
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You can trim before washing too, makes drying easier.
Keep an eye on it. Check his litter box or places where he pees, if the pee drains easily.