You need to only have one ammo loaded first, then fire, reload one and shoot right again. It appears it is faster than just spamming the trigger while full. It seems to cancel the "stop" animation between every shot. I might be wrong on this one though, but it's worth a try.
Hope this will help somehow !
The pump action shotgun, dumping your upgrades into reload speed is actual faster if you choose to just load a single bullet at a time and fire. Kinda neat
Tested this myself, with the m1897 pump shotgun you get at the start of the game this does nothing. Only works for the W870 TAC. You can cancel the pump while spamming m1 by holding m2, not sure if it makes it any faster but does end the animation sooner.
EDIT: Aiming in while spamming m1 on the tac is the fastest way to fire the weapon, provided you can click fast enough to fire as soon as possible. I tested this all with the pump attachment on, if anyone has an unmodded tac to test this with please do.