The cooler isn't technically necessary, but yes, the soft sided cooler will work just fine. The overall idea behind the rest period is to let the meat cool gradually so the fibers relax and the juices are "re-absorbed". Think of it this way: if you put a wet sponge in a plastic bag and squeeze it in your fist, then slice the bag open, a large percentage of the water that was in the sponge is going to run out. If however, you squeeze it in your fist, but then let go of it, after a bit the water you squeezed out of the sponge inside the bag will make its way back into the sponge. You can then slice into the bag and the water will stay in the sponge. Make sense?
As for the temperature rising during the rest period, this is kind of a moving target. It all depends on the size of the piece of meat and the temperature in the pit. A large cut at a pit temp of 350° or so will likely climb 10-12 degrees as the higher heat in the outer part of the meat makes its way into the inner part of the meat. A small cut like your sirloin tip has less thermal mass, and your pit temp was probably in the 225°-250° range, so the internal temp wouldn't climb, as you saw. You were right on in letting it sit another 10 minutes when you saw the juices running out.
Sorry for the long winded explanation, but a lot of folks don't seem to be too clear on the importance of the rest period.