Those bases are Weaver style and you need Weaver Style rings to attach to it. I use the Quick Release. And I am not trying to sound like a tight wad (which I am) but I purchased some $16.95 steel quick release rings from Traditions Firearms in medium height. I mounted a Simmons Pro Diamond 4x32mm on my White Bison and like you, have a fine shooting rifle. So far the rings have held up well.

As for drilling out the stock. Mostly I believe that is done on the White Lightening Rifles. Batchief or Underclocked can better advise you in that area. I have never drilled mine, and see no reason to.

As for conical bullet.. White are a slip fit rifle. You want a conical that is around .540 in your case. Contact Bull Shop conicals and see what they have. Another conical that shoots well in my Whites that are inexpensive... are REAL conicals. Put 60 grains under them and they will really shoot well. I believe I have some of the 300 and 380 grain laying around here. I am sure they would shoot well.

After you set that conical on the muzzle.. use that short starter and really give the hard ones a good smack to drive them into the bore.