Interesting project. You might look into brett or wine yeasts. They tend to be more acid tolerant, as a rule, but they come with their own set of issues, too. If you really want to get this down into the 2.x pH range, force carbing may well save you a big headache.
Besides, if you mean to sweeten it, what are you going to use that the conditioning yeast won't just eat up? If you say enough lactose to make this beer sweet enough to be balanced with a 2.2 pH, you're going to end up with an IPA with a body the size of Texas, tart and heavily sweetened, and, presumably, heavily hopped. It might not be a bad beer, but I wouldn't dream of calling that an IPA any longer. It's something experimental.