The point I was making with my comment re the wooden one was that it was much much cheaper than the Janome one. I actually have never embroidery on a cap anyway and think I would try just by flattening out the peak and pinning in a normal frame anyway first. Just couldn't justify the cost of the Janome one as I wouldn't be doing many.
I've watched a few videos - including some on using industrial machines - about how to embroider them, and all, including the industrial ones show the peak being flattened in order to get the front of the cap to lie flat for embroidering. After embroidering you just reshape the peak and they all look fine.
"Life may not be the party we hoped for, but while we're here we should dance". Anon