My first thought, like the other poster's thought, was: Can you wash this? If so, wash it several times in a row, in a scent-free, dye-free, hypoallergenic detergent, and see if that makes a difference. If you cannot wash it, or washing makes no difference -- Well, he has had a rash for six months and that is a long time. I know it will be painful, but get rid of it. There is no gradual way to do it; the bunny needs to get "lost" all at once. He will cry and scream but you do have to do it. Eventually he will forget about it. I would not overplay this with him, or cry yourself; I'd do as you have done and make other toys available and be sure to distract him with games, outings, etc. for a few days too. But don't do a full-on funeral for bunny!
By the way, if he's that allergic to this one toy, I would ensure that all other soft toys you have are labled as hypoallergenic. There are makers that specialize in washable, hypoallergenic toys. I'm assuming here that you have checked other possible sources for the rash, such as allergy to his bedsheets, to the detergent you use to wash the sheets, or his own saliva if he drools while sleeping (some kids drool a lot, and get facial rashes from the irritation of their own saliva on their faces while they sleep. Really.). Do ensure that there are not other things going on as well as bunny, but I feel sure that bunny must be part of it.
I do feel bad for you and for him! My daughter is 10 and has had her same lovey since she was only 10 months old. She STILL sleeps with Kitty every single night and Kitty makes every single trip out of town. I hope he finds another lovey, but if he does not, be aware that not all kids do adopt them!