Carpet roll tube if the hook isnt too aggressive. if it is, then usps has these triangular "tube" boxes that work great, connect two together after youve covered the limbs in pipe insulation.
Get a 4X12X 60 or so box. I shrink wrap the bow. Put some kind of bumper on tips. If enough room you could slip thick cardboard or even 1/4 floor underlayment type wood on each side.
I ship in pvc and two end caps. It’s more expensive but you never worry about it being damaged at all. But that is with longbows. I’m sure it would work with recurves too
For expensive recurves I build a wooden box/frame out of 1x2 fir and screw on a Masonite bottom, then line it with foam, wrap with bubble wrap, pack with styrofoam peanuts or newspapers, then screw on the Masonite lid. Never had one that didn’t arrive in perfect condition.
For inexpensive recurves I wrap the bow in bubble wrap, make a box out of cardboard, use packing tape it hold it together, and ship. Never had an issue with mailing them wrapped like this either.
When I built glass bows, places I shipped them to in the Masonite box include most of the lower 48 states, Canada, Alaska, Turkey, Guam, Australia, France, Switzerland, Japan, Great Britain, and Germany.