I tried both suggestions this morning.
I can't swim without kicking. However, I can swim with a light dolphin kick, which I figure should have a similar effect on "forgetting" the wrong-sided kick. I'll try to gradually reduce the kick as I feel more balanced. I can't tell for sure without a video, but, based on the relative ease of breathing on my worse side, I think my core twisted less this way.
Skate switch with focus on the bottom leg kicking as the top arm's hand reaches my face seemed to go well. No problem with 3-4 switches in a row (after which I need to take a breath anyway).
Another thing I realized might be helpful is vertical kicking with quarter-turns. I normally can't practice vertical kicking because the pool in which I usually swim is not deep enough. But today I had to go to a different pool, which has a true deep end. So I tried to coordinate each quarter-turn with a stronger kick from the leg in whose direction I was turning. This was easy to do and felt completely natural.