I would cook them and go ahead and pull them and put them in vacuum sealed bags and get them in ice or the freezer to get them cooled down as quickly as possible.
Freezers don't usually cool quickly enough. They are not designed for cooling. They are designed for keeping already frozen food frozen, or freezing already cool foods.
I completely agree that getting the meat cooled as quickly as possible is of utmost importance.
Handling the meat can transfer pathogenic bacteria (even if gloved). A particular Staphylococcus bacterium (found in the environment and on skin - 30% of people are estimated as natural carriers) can be nasty. These bacteria are often responsible for illnesses stemming from cooked foods. They require warmth and time to outgrow to a particular point at which they produce a heat-stable toxin (a toxin not affected by typical reheating temperatures).
Pull the pork while still hot or warm and either place
shallowly in pre-chilled metal pans, placing the pans in the fridge, once cool, to chill; or place the pulled meat in Ziploc or FoodSaver bags - do not pack the meat tightly - then plunge into a large pot of ice water, submerging the meat but keeping the top of the bag open (to release heat) and above the ice/water level (so as not to get water in the bags).
If using the first method, transfer the chilled meat from the pans in the fridge to fridge containers or bags and fridge or freeze as desired. If using the second suggestion, press the air out of the bags (or use your vac-sealer), seal and fridge or freeze.