How do I politely decline student requests to meet during my "research time?"
I've yet to come up with a nice response that I'm satisfied both:

Reminds students I'm busy with other work (despite the fact that their tuition pays part of my salary).


There is no need to &quot;remind&quot; students of anything. They likely neither know nor spend any meaningful amount of their time wondering what you are up to when you're not teaching them. Nor is it any business of theirs where your salary comes from.


that still encourages them to keep reaching out for help and encourages them to think of me as the approachable professor I try to be (at least on Monday, Wednesday, and Friday).


It is commendable that you want to be approachable, and to be seen as approachable, on Mondays, Wednesdays and Fridays. However, from the students' point of view it is likely to be seen as inconsistent that you are so approachable on those days and yet refuse to meet them on other days. In my opinion, any attempt to explain the logical reasons for this inconsistency is going to have very limited (if any) effectiveness -- the students simply don't have a good enough understanding of what the life of a professor (or any professional of a similar age and career status) looks like to be receptive to your explanation.
The upshot of this analysis is that you need to accept that setting the boundaries that you need to set to get your distraction-free time for research is going to make you look slightly less approachable and/or likeable in the eyes of the students. The problem is not one of finding the right words to put in an email, but instead of accepting that &quot;nice&quot; and &quot;approachable&quot; is sometimes inconsistent with &quot;gets things done&quot;.

Suggestions for a nice email response?

Here is my suggestion:

Dear [name of student],
I am afraid I am not available to meet with you tomorrow. As I mentioned in class and in the syllabus, my office hours are [insert office hours], and I may have limited availability to meet at other times on Mondays, Wednesdays and Fridays, but I am not available on Tuesdays and Thursdays. If you have an urgent need for help, feel free to try talking to [name of TA] or to a fellow student. I do very much appreciate that you are seeking help to improve your understanding of the material, which is a positive thing that sadly not enough students do, and look forward to seeing you at my office hours sometime soon.
Best,
[your name]