Those White Mountain Carbines had a short barrel, so large powder charges will do you little good. I would not use anything over 100 grains. 80 would be where I would look.
I would look into some conicals or sabots. If I wanted to shoot sabots, then I would look at a 300 grain XTP in a green harvester sabot. Get the .430 diameter. Try them. Another thing you could try is the 250 grain Shockwaves.
As for conicals, there are maxiball, great plains conicals, and if you know the bore size, BullShop will custom size them for you to the barrel. They should shoot good in around a 460 grain. And that with 80 grains will stop anything you want to shoot at.
Like they said, with that fast twist, roundballs would have to be shot with a very light powder charge for any kind of accuracy. And then you will not get much distance.
Look at the different conicals you see in adds. See if one appeals to you.