The main issue is the depth of the new switch. With a box depth of only 2.75", there's not much room for wire. I'll guess i'll try the deeper box.Three 14-2 cables is 6 + 1 for the ground + 2 for the switch for a total of 9. 9*2 = 18 cubic inch capacity minimum.
You may need to dress the conductors to make this fit. Maybe a pic would help us see what you are dealing with.