Ken2222, I'll try to explain how I would tackle the bolt bore. (for what that's worth..

) I agree 100% about using the lathe. It will be much easier to attach the sliding jig you will make and use to pull this off. The center line of the lathe bed is true center to the boring you will do.
The first thing I would do, as you have mentioned is cut the new rear stock portion long. I would also make my temporary cut square to the bolt bore so the bit will enter the stock easily, as opposed to trying to bore into an angled cut and risk the bit walking. If you dont feel you have the extra length to do this, I'd suggest you cut it at any angle you can, then glue a sacrificial block to the stock that can be cut square to the bolt bore. To determine square to the bolt bore I'd remove the existing rear stock from the receiver, and insert a long dowel rod through the bolt hole. Now lay the old stock with dowel still in it, on the new rear stock, and mark the edge of the dowel (basically trace the edge to make a line) onto the stock with a pencil. You can then use a small square to mark your cut.
Now I would chuck up the long bit into the lathe. Next slide the old stock onto the bit, all of the way, standing on edge, and bottom of stock up.
I'd use 3/4 birch ply to build my jig, starting with a bottom plate of the same width as your lathe bed. Then apply ears on each side, ( the length of the jig base) hanging down on each side of the lathe bed... This is the base for the sliding jig, and it should slide easily on the lathe bed without slop. The length of this base will basically be an inch or so longer than the rear stock.
Now you have 2 areas of your stocks that will basically be the same, pistol grip, and the very end of the butt. With the old stock still scewered on the bit, finish building your jig to completely grip the old stock. I'd suggest you build four " L" shaped pieces with a 45 degree block glued to hold the " L"s at 90 degrees. Now attach the four "L" brackets to you sliding jig base in a fashion that sandwiches the sides of the stock front and rear. I'd also keep the two rear " L" brackets flush to the rear of the sliding base, then conect the to brackets to form a solid rest against the butt. This will keep the stock steady in the jig. You will also have to add blocks to the jig to hold the stock at proper height.
When you install the new stock in your jig you may find the jig too tight or loose. Simply make the adjustments equally to the " L " bracks to remain centered. Once the new stock is in the jig firmly, I'd use some cedar shims, here and there to stiffen things between the stock and jig.
Now your ready, turn on the lathe, and slide the stocked jig towards the bit to bore.
Your on your own fitting to the receiver...

I sure hope this helps, and I'll try to answer any questions if I have you confused.
Keep us posted and good luck, Ridge