I've been able to get the Espurr BEFORE Narcissa's Gym in both V12 and in V13, so you needn't wait on that to get it. But here're some tips:
✔ Strawberry ice cream is its favourite flavour. Don't bother with the others, even the Blue Moon Ice cream; just go with strawberry.
✔ I've been able to "speed" befriend Espurr by just chilling with it on the swings constantly; there're two "dialogues" that lead to a "bonded a lot", but, regardless, just spam "yes" to whatever Espurr asks.
✔ Be careful when walking with it; it gets in your way quite a bit, but, in V13...DO NOT try making it go on the slide while walking with it; it's bugged and crashes the game - I've tried and Espurr always "flies away" afterwards. I'd report it, but I don't know if it HAS been reported already.
✔ This might just be in my experience, but trying to leave Espurr after doing some of those things for a few minutes for the third time is when Espurr decides to join you. Therefore, SAVE before you leave for the third time; that way, you can reset for the Espurr you want.
Hope this helps.