but does Inyfo FS care if you start at Cottonwood Lakes TH, but cross over to Cottonwood Pass, and then come back to Cottonwood Lakes TH via Cottonwood Lakes?
I am POSITIVE that if you actually did that, they would not care. They only care which trailhead you start from, nothing else.
I am also positive that a lot of people get the two trailheads confused, and end up on the wrong one, or get the permit for the wrong one.
As I mentioned, I have started at C Pass, then came out from the lakes, and cut over to the Pass trail....no reason why you could not do that at the beginning. I think you lose about 90 feet if you were to drop to the trailhead, but the CL trail heads due west, then heads due north. At that point if you kept heading west, you'd run into the CP trail, just by dropping to 10,000 feet. In fact, this might be quite fun, inasmuch as there appears to have been an old trail exactly doing this, and trails don't disappear. They just hide the beginning and end of them, but the trail itself is still there, and it'd be fun to find and follow it! You can't really get lost. There is also a spur trail (unmarked) on the CL trail that takes off downhill to the packer campground, don't mistake that for the old trail.
One other side note: on the PCT from Chickenspring Lake to the intersection with Soldier Lake trail, there MAY be a spring. On the map, to the west of Cirque Peak, there is a section of trail where it reverses direction, and heads SOUTH for a short distance, on some maps showing it going around a small lake. There is no longer a lake there, but there is often a creek. When you are travelling southbound, about half way, you reach a somewhat large cut log face that defines a path to the left downhill into some large boulders where the creek resides (and a nice small camp area). I wouldn't depend upon it for water, but if there, a nice place for a break.