Child sleeping in bike trailer
I hacked the seat in our burley trailer to recline by decoupling the seat top from the frame tube it ran through (required disassembling the trailer), and suspending the seat top instead from some climbing webbing which could be tightened or loosened with a typical adjustable nylon clip. I did that so the kid could recline a bit more and sleep more easily. I also ran some pool noodle down the sides of the interior to act as bumpers. The first thing made the kid more comfortable. The second thing made me more comfortable. She'd still fall asleep and her head would sink or flop, and I couldn't believe her neck didn't hurt, or that she wasn't choking herself with her head that far forward. (I eventually tried that position and realized it does the opposite with the chinstrap.)

Finally, I learned to chill.

The same thing happens to kids in cars. And on airplanes. And in the jogging stroller. And when you walk a baby around in a snugly. Are you aware of actual injuries resulting from this? I'm not. And I looked. I couldn't find anything on this (or in car seats) on pubmed.

Don't get me wrong, I used to worry about this with my kid in the trailer as well. (And I took mine on 50-80-mile days beginning when she was 9 months old.) However, before I did that, I created a baby test dummy (a large doll, weighted, with a 3d accelerometer on its head. Yeah, I was crazy.) I ran the numbers from the bouncing in the trailer against numbers from bouncing walking around the house, down the stairs, on the trail nearby, and the peak Gs were all comparable, and very low. When she was older, she never reported a sore neck from sleepflopping in either a car or bike trailer.

There are gadgets designed to support kids heads by wrapping around their foreheads, but frankly, I'd worry more about a choking risk from that than I would a neck injury from a floppy head.