I'm glad you're having fun with TaiFab!
For both the alligator and the last Fish Power, it sounds like you might be missing a crucial item. Have you searched the bathtub in the inn? :)
Viewing post in Taiyaki Fabulous Museum of Fish comments
Ooh thank you! That was actually the missing clue. I’m sure I was there before, but spammed A instead of going left and down and assumed it was just decoration like all the other grey objects. It’s a bit confusing where you press A and where you move into it but only from one direction, where the sides have collision, like a door. Now I blew through the rest and summoned Chad. I only felt stuck for one more moment, when I had to understand I should re-visit two locations of a previous quest and to find the remaining ten puzzles because now it would do something different than before. But I admit I did not really read the text for that quest, I was just sprinting for the next puzzle ^^ Thank you so much for this enjoyable experience!