Alright, I think I figured out how it's going to work. I found this formula that can reference any date and return the Monday for that week, so I put that in the top left cell of the linear calendar:
Every other cell just references that one, and adds a +1 that increments for each column or a +3 if going from Friday to Monday. I cheated having to type +2, +3, +4, +5, etc. etc. by using the COLUMNS function to count $A$1:A1, and then had the fill handle increment it for me. I also found out that you can hit CTRL + J while you're setting up custom formatting to enter a soft return, so I found a conditional formatting rule that identifies the first working day of any month, and set it up to change the formatting to mmm↵d:
Now everything seems to be working like it should!
And I totally see what you meant about calculating the Student Attendance column, it's a lot trickier than it seems. I ended up calculating all of the non attendance days for each row, then subtracting that from a COUNT formula and it's been doing good so far. But only time will tell!