Just an opinion, but I wouldn't want the sight screwed to the action at one end of its base and to the stock at the other end.
Those type sights are quite high and if bumped or knocked can apply a fair amount of leverage to their base so I would want it attached as strong as possible.
If it was my rifle I think I'd be fitting a tang to the rear of the action so that the sight base is held by two screws which fit into threaded holes in the tang.
The tang would need to be a permanent part of the action and therefore may require fitting by welding and would also need to be secured to the stock with a woodscrew at its rearend.

It's my opinion that sights of any kind need to be mounted as solidly as possible and especially so with tall adjustable peeps like the one in your photo.

So yeah, as you say, add tang and drill/tap both ends would be best IMO.
Great project by the way.
Best of luck and please keep us all informed as you progress.