Yes, absolutely use separate brass for each. With that much difference between rifles (NOT uncommon), you would be over working your brass anyway, and brass life would have been short.

You might also think about neck sizing your Lapua brass in your Savage. Brass life will be longer, and usually, groups will be smaller. Then only FL size the brass when it becomes hard to close the bolt again. You don't want to over do it with bumping the shoulder back, just about 1/8 turn of the FL die at a time until the bolt just closes on the case should do it.

You don't need to buy a bunch of "thingy's" or "trinkets" to do that. The "old school" way of turning the die and checking it as you go will work.

Just my 2 cents worth...