As @toowonderful mentioned, I think it is very important to write thank you’s for teachers that take the time to write the recommendations. It is one of many extras they do for the students. We did the same for the GC - we were fortunate to be at a school where a close relationship for student and parent was easy to be had with the GC.
We bundled the thank you card with a gift and gave it just before winter break - basically serving as a cross between a thank you/Christmas present, and kids wrote something along the lines of, “regardless of how my college decisions turn out, I want to thank you for writing my recommendation…” etc. When doing it at this time, the thank you is made independent of the results.
I would be disappointed it the favorite for a recommendation couldn’t do it like OP faces, but sometimes the best recommendations come from someone that sees your students beyond their grades, and really likes them as a person. Maybe they didn’t get the top grade but they can write about the student being the most inquisitive, broad thinking or they participated a lot - things like that. I think it can be good to consider teachers outside of a student’s best grade.
On the other hand, perhaps your student could tell his current math teacher that he understands she is super busy and can’t write the rec, but perhaps she could mention to the Pre-calc teacher how well he is doing now or give some kind of feedback to them to include. That way the pre-calc teacher would have something to add like his growth or upward trajectory in his math studies. Just a thought.