The use of a small base die with a pump action rifle has nothing to do with precision, but can be an advantage for reliable feeding and chambering.

Since there's no camming leverage to help close a pump gun on a slightly oversized cartridge as there is for a bolt rifle, small base dies offer an advantage. If a guy has a sufficiently loose chamber, then a standard die will size the base small enough to close comfortably. Some guys note in the 7600's, it can be almost impossible to close the bolt slowly, as in sneaking it closed when chambering that first round upon arrival at the stand, when using a standard FL die. Slam the pump forward and a guy might find little difference. Similarly, in 223/5.56 for AR's or 30-06 for M1 Garands or 308win for M1a's, if you ride the charger forward slowly, match chambers in custom barrels might argue with reliable chambering of rounds sized with FL dies.

It's a common enough issue for pump guns.

For the OP, I'd personally measure the case bases of once fired and FL resized rounds from your pump gun. If you have sufficient downsizing of the base for reliable feeding, don't look for a problem where there is none. If there's insufficient base sizing with a standard FL die, then I'd measure the same offset for your bolt gun, compared to the FL resized base AND the small base resized base. If the small base isn't significantly undersized, roll on and be happy, small base everything. If the small base IS significantly undersized, then I'd compare the headspace of the fired brass from each rifle. If a significant shoulder bump in the pump gun would be a lesser but sufficient shoulder kiss in the bolt gun, then I'd roll on and small base everything. If the headspace dimensions don't match up well, I'd continue to use separate dies for each.

Important in all of these comparisons is to observe the appropriate base to ogive length for each chamber. If you have a short throat and close leade in one rifle but not in the other, then you can be sure you'll struggle to find a seating depth which really well suits both rifles, regardless of the results of the above comparisons, so you'd need to seat the bullets differently for both rifles. If the throats do match up for base to ogive length AND the above criteria are met for headspace and base size, then roll on with one load made for both chambers. Considering a bolt gun against a pump gun, the likelihood of this all lining up might be on par with finding a rainbow farting unicorn.