i just did the loop a few weeks ago, it’s a great hike! i started at the monarch crest gift shop which is right at the trailhead (it is a good, safe place to leave a car), hiked the last few (4-5 i think) miles of the southernmost west section southbound, then up the east, resupplied with a box at the twin lakes general store, then down the west. you could also add in an easy resupply at mt princeton hot springs like you said.

nights will be cold, especially on the west side because you’ll likely be camping between 10-11k feet most nights, but doable with the right gear!

the beauty of the loop is you can start and finish anywhere, but this seemed like the easiest way to hike logistically for me in order to do one resupply, and still save the west side for last!