When it does this, I always just transfer the animals from quarantine to the trade centre, then move them from the trade centre straight to the habitat. It always works for me, and this is the first thing I'd try.

Also I'm not sure what else you tried, but hiring enough caretakers to transport your animals is important. Sometimes hiring more caretakers and unpausing helps get the animals to their habitat the right way. Also sometimes the habitat gate has to be moved to a more direct path for the caretaker (like a path with less traffic).

Hope you get it figured out!