I don't think it is possible to get a full clear with the current rules of the moogle stamp bingo.
So, lets set up the sample space. We have a 5x5 grid; the grid is cleared if 3 horizontal, vertical or diagonal lines of 5 are completed; and at most in 1 turn you can stamp 3 spots.
A fully completed bingo/mog stamp board has 12 lines completed. (5 horizontal lines, 5 vertical lines, and 2 diagonal lines.)
The "best" removal will be the center spot, breaking 4 completed lines. (a vertical, horizontal, and both diagonals)
We now have 8 lines completed.
The next best removal can be in many places and would be breaking a horizontal and vertical line.
We now have 6 completed lines.
The next best removal is the same as the last, and would be breaking a horizontal and vertical line.
We would now have 4 completed lines.
For it to be feasible to get a full clear with the current rules, it would have to be possible to get it down to 2 completed lines by removing 3 stamps, but it isn't.
Though if there was ever a case where you had a symbol that represented 4 spots, a full clear would be possible.