When making a 45 degree cut, the longer edge will extend 1.414 *
Thickness / 2 further than the short edge. In your drawing, the short
side was 6 feet long. There are two 45 degree cuts, adding 1.414 *
1.25 / 2 = 0.884 inches to each end, so the total length ends up being
6' 1.77" on the outside.

The much easier way to do this is to cut one side at 45 degrees and
just mark exactly 6' on the back of the board (or better yet, hold the
board up to the wall and mark the length EXACTLY). Then cut from the
front side being very conservative and leaving yourself some extra
length. Then flip the board over and see how close you got to the
pencil line. Flip it back to the front side and cut off a little more.
Repeat until you get to the exact correct length on the back.

Also, assuming you have a 10" miter saw (and it IS compound if the
blade bevels, even in only one direction), you should be able to stand
the molding up on edge and, therefore, only have to rotate the table,
not change the bevel. You should always keep the front side of the
molding facing outwards (towards you), since there will likely be some
chipout on the back. If you'd prefer to keep the board flat and tilt
the blade to 45 degrees, you should make all of your cuts with the
outside facing up, again to isolate chipout to the back side.