When loads are developed - they use x powder and x primers and x cases and x bullets.

Ok

Now will Federal or CCI or Winchester Primers cause you problems?

Probably not, but what you have to do - as a good reloader is look at your loads and test them before you run a production run.

Load 5 rounds and take them to the range and try them out and see how accurate they are compared to what you were using.
See if it exhibits excessive chamber pressure.
See if it shoots as well as what you were using.

If you never reloaded with that load before, you should start out about 1/2 a grain below max loads and work your way upto what works best for you.
Just make sure to check the primer pockets and look at the cases for chamber pressure signs.
If the head of the case expands more than .001 then you have too much chamber pressure. You measure it with a blade mic or your calipers that you use to check oal. Just measure inside the extractor groove.

As for bullets, what ever works for deer will also work for bears.

At ranges less than 100 yards, most all bullets will do a good job on a bear. It's bullet placement that counts, not the manufacturer that kills bears.

Most bullets will fly the same at that range as long as they are ofthe same design, so what you could do is load with Hornady Spire Point 140 gr bullets for deer and then use Nosler Partitions or Barnes X Bullets or Trophy Bonded Bear claw bullets for bears.

As long as you use the same primers and the same powder and the same amount of powder and the same weight of bullets and the same pointed nose bullets and the same cases.

They should hit the paper in the same place and the rifle should not have to be zero'd in between hunts.

Thats the good thing about reloading, you can control what goes into your ammo.