Hah, I am doing the same thing on my chess site [1]. On my site you play against an opening repository so there is not even any real calculation done just a lookup into the repository what the next move should be. It feels really weird having the computer respond when you feel like you haven't even finished your move. That is why I added a 300ms delay [2] to make it feel more natural.
[1] https://listudy.org/en
[2] https://github.com/ArneVogel/listudy/blob/master/assets/js/s...