Fortunately, our late rains have left abundant water in creeks, springs, and tinajas across the park. As such, there are literally too many water sources to describe in the text of this response. If you have the BBRSP Exploration Map, you will find the springs marked with a little blue squiggly symbol, and most of the marked springs should have at least a little water right now. It has been a few years since I have traversed the Lower Shutup, but there was water in the channel in numerous locations, as well as in the deep tinaja below the house sized rock that obstructs the drainage about 1.5 miles from the Lower Shutup Trailhead. The most dependable water in the Solitario is at Tres Papalotes, though be advised that we have had many recent bear sightings in that area as well.
My greatest concern about this proposed route is the Lower Shutup itself. The LSU is a tricky and dangerous place. I've never gotten a satellite phone to function within its confines, and GPS signal is sketchy and inaccurate at best. There are also a few weird, tricky moves that one has to do to cross it, the most dangerous being that house-sized rock. If anything goes pear-shaped inside the LSU, there are no comms, no access, no place to land a helicopter, and no place to extract a wheeled litter. When you are in the LSU, you are on your own in every sense of the phrase. Remember where you are at all times and make good decisions.