Well, this actually turned out quite nicely, although I am not sure I would try it again, too many difficult to control variables. This was truly a case of not having the right tool for the job. This is not a recipe and you should not necessarily try this at home, but here is what I did. I put wood charcoal on one side of the grill and tried to get a nice bed of coals going. On the other side, I put some foil under the grate to catch drippings and the idea was to get the pork as far from the direct heat source as possible.
I only have an oven thermometer, so had to open the lid to check temperature. My goal was 225 degrees F. It would have been better to have a thermometer I could have stuck through a vent. In any case, once I had the temp to 225, I put a nice rub on the pork, put it on the grill and threw in a few mesquite wood chips. I checked temperature often and added some charcoal at one point. Finally, the temp dropped to a level where I could no longer get it higher, even with playing with the vents (my grill is a cheap one and the vents are half rusted).
But, when I took off the pork after a couple of hours, it was done and very juicy, with a nice smoke flavor. Not sure I’d use mesquite again, but it was good. I chopped it and put a thin vinegar-ketchup (Carolina) sauce I had made on it. It was really quite good. Little Buddy would have eaten as much as she could if I had let her. Humans liked it, too.