This sounds like a really interesting project. I think I would applique parts of the caps to a larger block of lighter weight material for ease of and consistent weight piecing (i.e. you wouldn't be sewing the cap into the seams.)
As caps vary you have some interesting options. I think I would just use the logos as applique pieces in one cap's block. You could use the main panel with the logo and trim it in the cap's secondary color(s) as applicable. Include pieces of the brim and the back of the cap if they have logos.
Select the hats you want to use, decide on the layout of each applique set and go for it!
Have fun....