If I were to do that. First I would take a correct sized pilot tipped drill bit & go in from the removed triggerplate area & go up just to the tang & just touch it to mark it.
Then take barrel & tang out & centerpunch tang good, but Just barely towards the muzzle, as it is going to try to walk towards the buttpate most of the time.
Then I would take Epoxy & fill up a Pilot tip drill bit all the way to about 1/8" of the tip. Let it dry, chuck it in the drill press & carefully take the epoxy down to the steel by filing as you rotate it in the drill press. This gives you a solid shaft & it won't eat into the wood as easily & oblong the through hole.
Hopefully you do have one good clean hole thru the stock. Put the snug filled Pilot tip drill bit in & Drill it with Very Little pressure & put it out several times to remove the chips.
Then cut the countersink from the top side.
I have done this several times on rifles I have rebarreled & it worked for me.
However, if you decide to do it ALL over, fill the stock as well with a glued in dowel & redrill all of it.
Keith Lisle