There is old fashioned method of carrying game birds, it may not work for rabbits as they have small feet. Take a piece of wood perhaps 2 x 12 x 0.25 inches. The exact dimensions are not important but will depend on the type and size of the birds. Cut a slit through the middle of the 2x12 face, about 0.25 inch wide, stopping about 1in from the ends. Enlarge one end of the slit to have a circular hole big enough for the bird's feet to pass. Affix a 15 inch strap, as a handle, to the ends. Add some way of blocking the hole to stop the feet sliding along the slit an out through the hole.
The wood should match this crude diagram:
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This picture shows a version with two slits, each has a hinged metal plate to block the hole:
