I also have a number of 2.375" boards and the Mechanics Institute set will play fine on it. In fact I know fellow that has owned one for over 6 years and he has only used a 2.375" square board.
However, I certainly would not use a 2.25" board size for this set. The King base diameter is 1.875" which would make the percentage on a 2.25" square as, 83%, way too high.
On a 2.375" square that is 79% which is about as high as you would want it to be. The 2.5" square size ratio would be 75%, which is just right by pretty much all standards used. Now, the 2.75" would give you a 68% ratio which, as cgrau said, overkill. Plus the 2.75" boards are really quite large overall size, and might be too big for most tables.
However, I see that HoS sells a number of boards with 2.75" square size. As long as you're ordering the Mechanics Institute set, a 2.75" square vinyl board is only about $8. You could order one, along with a 2.5" and a 2.375" one and see what you like. Of course whatever you get, please post pictures!
That Mechanic's Set, which I too have had my eye on, definitely needs 2.5" squares, which is a major drawback as I own nothing but two point two fives. -Gg