First off, if you didn't strip the stock, with a serious paint, or epoxy, stripper, the shoe dye will remain on the surface, and wipe of with only a little handling, since most Italian guns have a sealer on them that is nearly bulletproof. A hard finish will protect it for a while, but quite soon that to will wear off too, and expose the colorant to wear.
For an oil finish I like linseed oil,( for one stock I buy the good stuff sold at art supply stores in small bottles) mixed with a little turpentine, and applied hot,and heavy, until the wood won't take any more. I then wipe it down and put it up for a week or more (depending on the ambient air temperature, and humidity)I then hand rub in a mixture of linseed oil, turpentine, and bees wax, all brewed up together hot and then left to cool into a soft paste.
Hungry Horse