You either have a Cottonwood Lakes, Cottonwood Pass or Trail Pass Trailhead permit. The powers that be do care where you start. I've run that one by the folks at ESIAVC.
Basically, if you have CL Trailhead permit they want your first night's camp at that can be gotten to from that trail. In other words, how would you get to Chicken Spring Lakes from the Cottonwood Lakes Trail other than use the Cottonwood Pass Trail for most of your trip.
Fortunately, you can probably exchange your permit when you pick it up for the trailhead of choice. Both trailheads have big quotas.