Well, because someone determined it had a 1.540" pin height, I think boring to 4.060" is unavoidable. Standard pin height is 1.560" and some of the better selections are 1.563 ish.
The top ring height of a more desirable piston will pass the old ring ridge. Even if it is only a very slight one it will be undo-able.
A good set of heads will allow enough power out of a 350 to surprise you, or from here you could buy a complete rotating assembly and go 388 I think. Find out the head and block numbers. Going to the shop; looking forward to hearing the rest.