Yeh, my male is quite happy to "do the deed" with his little round bed. Unfortunately, he finds himself in the same position that your pup did. To get a bit clinical, I think when the hairs on the tip of his sheath get too long, they get pulled back and into the sheath, and the rest of him gets stuck ...
My (adult) daughter usually gets stuck playing Nurse. It seems like AstroGlide has more than one use