Hey does your DH like doing the dishes? If not then your argument should at least be a little easier. Lol but if it's this hard to convince him just to have the facilities for future use I don't envy you in trying to convince him to actually get a dish washer!
Steve has a great point there, I think it's a good one to put forward to your husband to convince him. And $350 sounds pretty reasonable to me.
Rubber gloves aside, I know if I happen to cut my finger even just a bit, or accidentally burn it on the wood heater, it does make dish washing in the hot water unpleasant. So to me having the facilities for a dish washer particularly in case of something more serious is a great idea, and considering you're making the place wheel chair friendly, I think you've found your main argument "for" the case, particularly if he dislikes washing up