Crawfish reminded me.. it is called wood glue for a reason.) Okay. So I left the surgical implants in a bit long so that they could become pegs to insert in the base. Then I made a little template out of some scrap clementine box wood and drilled 1/8" holes so I could line up the pegs with the holes.
Ultimately, this ended up improving the toy for me. On the next one I built, I pegged all the animals to the base using the same method and it was very easy and resulted in a stronger toy.
Necessity is the mother of invention but I guess failure is the father of improvement.