FWIW, Winchester 1200 (pump), 1300 (pump), 140 (auto), 1400 (auto), 1500 (auto ) & Sears 200 (pump) shotguns all used the same "family" of receiver & buttstocks, etc , just as Remington does with the Model 870, 1100 & 1187's - so there's plenty of interchangeability there.
While a PG on a short barreled shotgun (aka: Cruiser) make a nice "Whippet" gun (as in: Whip it out"), in reality it makes them hard to control in 12ga, and marginal in 20ga.
I would suggest you shorten your existing barrel (if it's a plain barrel) and try it that way, before opting for the PG kit.
If you have a VR barrel, since shortening can be problematic for an inexperienced fiddler, there are usually short/riot barrel for the 1200/1300 online at one or another of the barrel dealers (Corsonsbarrels, the barrelexchange, etc)
carries an ATI heat shield for $21.50: