Oven – Braising with 2 dutch ovens – time

Sadly no mater what you do, you are going to come out with non-optimal results. I would go with the oven method over the grill as you need to get the meat up, but you want to do it with out losing all the juice/fat you have already taken all the trouble to convert to gelatin. The grill is going to raise the outside of the meat much quicker that the inside (mostly, you can use your grill as an oven but in this case I don't think it brings anything to the table that you regular oven doesn't).

The best way, and by best, I mean the one with the tastiest results, is going to be put the butt in a 200 to 250 degree oven and let it come up to temp. This method is going to take the longest but will preserve the meat that come off the grill as close as you could.

Or, you can heat just like a roast and let the outside overcook a bit.

or, depending on how you are serving the finished product, you can slice cold and finish in a skillet to heat though... this would be great if you are doing sandwiches or pulled pork.

If you are feeling adventurous, 130 degrees plus smoke is a little on the rare side but it would still be delicious... not for this one as you let is cool on the counter top and fridged it but on the next one, I might just eat it then.

I'm not sure what you mean by a "stoneware pan" in this case...but some pottery used for cooking shouldn't be washed with soap.

If it's glazed, then you should certainly clean it with soap, there's no reason not to.

If it's unglazed, and it's really stoneware, it should be safe to wash it with soap; since stoneware is watertight, the soap will stay on the surface, and you'll be able to rinse it off, so using soap is fine.

For water-permeable baking stones, like a pizza baking stone, or some domes used for baking bread with a french-style crisp steamed crust, then using soap is a bad idea, because you typically can't rinse the soap out afterwards. I've never tried washing one with soap, but the instructions that came with my pizza baking stone say that a soapy pizza baking stone is ruined and unusable for food.

For unglazed earthenware used for pizza, bread, or similar, you clean it with water and/or heat: if it gets dirty with something you can't wash off without soap, the remaining option is to burn it off by heating it to a high temperature in the oven, and then brushing or rinsing off the ash. This removes the "food" from derobert's list of things needed for bacteria to grow, by turing it into less nutritious ash.