The White Mountain Carbine as I remember had a 21 inch barrel and a 1:20 twist. So I would suggest some large conicals try with 70 grains of Triple Se7en and say a 385 grain Hornady Great Plains, or perhaps a T/C 370 grain Maxi Ball. It should also shoot sabots once you find the right one. It sounds like an exciting rifle to own. They are hard to come by because normally those that have them keep them and those that sell them regret it. You can also get a Green Mountain Drop in barrel for that rifle.. It has a 1:28 twist I believe it is..

Good luck with the rifle