Tell your professor of your plan to apply to 16 schools, and let them decide if it’s too much. They are capable of making their own decisions without you doing that on their behalf. It’s nice of you to worry about the professor’s well-being, but unnecessary, and counterproductive if it ends up undermining your own success.
And as for the professor regretting having to send so many letters, well, yes, I think that’s something that many of us experienced. It’s unconscionable and stupid that the system works that way. But that’s not your fault. If it’s really a lot of schools and a big time sink for the professor, perhaps they can arrange to delegate the task of uploading the letter to a clerical staff member at their department. That’s their concern, not yours.
Edit: I see that several of the answers and comments are discussing whether the number of schools you’re applying to is appropriate, not from the point of view of your specific question about the professor’s letter workload, but in the sense of whether it’s generally in your interest to be applying to that number of programs.
It seems to me worth acknowledging that this sort of gratuitous advice has nothing to do with your actual question and was not solicited by you in any way. In particular, there is no need for you to explain the reasoning that led you to that number or what mix of schools you’ll be applying to, either to the people here or to the professor you’re asking for a letter from (although if you want to seek advice about such things, it’s certainly fine to discuss it).