That is a good thought. I know you can get Baltic Birch in 1/8" thickness and if I recall, 1/16" is available.
Yep, typically called "aircraft plywood" (I think because it is used for model aircraft) I used it for Rockets.
The thinnest you can buy is 1/64". But that stuff comes in small sizes, like 3" x 24" or some such. (it's been a while)
I know you can get aircraft ply in large sheets, but you might need to look to a hobby source.