What tool or technique is best for cutting circles in matte board?
A hole saw should work fine if you can find a fine tooth model. They are not as easy to find as the common coarse teeth models meant for wood. One with a thin kerf would be best.
I have also used the sharpened steel tubing trick mentioned by @fred_dot_u. An alternative to punching is pressing the sharpened tube against the board and spinning it. The spinning action makes it work like a circular knife, cutting the fibers cleanly. If you can't find a 35 mm steel tube, you can get almost as good a result by wrapping thin aluminum from a soda can around a 35 mm thick piece of plastic or wood.
To align the holes perfectly, it will help to make a template. Cut three holes in a smaller piece of board, align two of them to existing holes on your workpiece and then put your cutting / drilling tool in the third hole to keep it in correct position.