Wow, that must be some mite infestation. =\
That being said, I would not advise using that dip. I’m surprised the vet would suggest it (that more so than the Dawn). Here is what MediRabbit has to say:
The use of over the counter products against mite or ear mites should be avoided. Most contain pyrethrin. While one of the least toxic insecticides for animals and rabbits, pyrethrins/pyrethroids are neurotoxins which lead to continuous “firing” of nerves. Observed problems in rabbits include limb paralysis, seizure, coma, tremor, sometimes death.
Many of those products contain additives that prolong the effect of the pyrethrins, and inhibit the breakdown of these products in the body. They can lead to toxic reactions in rabbits. Piperonyl butoxide is one of them. If no immediate death, there should be **slow** recovery with time, around 48 h.
If you absolutely must bathe the bunny (I’m not familiar with mouse mites), then I would not use Dawn. I would use something like Zymox, which is safe for rabbits and will help prevent infection if bunny has been scratching their skin too much. But, really, I’m doubting you would need to take such extreme measures. I would only see this being appropriate in an extreme case. From what you’re describing, it sounds like that this is far from being an extreme case.
As the others said, Ivermectin and Revolution should be effective in treating mites. It might take more than one treatment though. But I would definitely start there before resorting to bathing.
Just so you know, anything with fipronil should not be used on rabbits. I remember a very sad case on here where a vet used Frontline on a rabbit, insisting it was better for mites. While it is good at killing mites, it’s also good at killing rabbits. It’s highly toxic to them. The story was heartbreaking. So please make sure your vet knows that.