The transfer buttons are typically used in conjunction with a drill jigs that make sure the holes have the same axis. For irregular pieces like the OP's, only drilling oversize holes in the two halves marked by buttons will be sure to allow alignment, and then you need a filled epoxy to fill the gaps around the dowels.
The better way is to follow Scorch's approach. I will re-describe it just in case reading it in different words helps someone.
1. Remove the stock from the gun. This will expose the wood normally inserted into the back of the receiver. All drilling needs to start into a wood surface that is concealed when the gun is assembled, and taking the stock off the gun is the only way to access those surfaces.
2. Glue the two pieces together, do all required wiggling and adjusting to achieve good mating so as to make the glue line as invisible as possible. Clamp the parts so they can't slip. Rubber bands are your friend for this with irregular shapes. Get all excess squeezed-out glue wiped clean, and removed by solvent (alcohol works with wet epoxy) if necessary.
3. Let the glue set, but before it fully cures, you want to drill your dowel holes. Note that the dowels don't have to be perpendicular to the glue plane. They can go diagonally across the glue plane. They just have to start from the normally hidden part of the stock, pass through the glue plane and not go deep enough to emerge from the finish on the other side.
4. Glue your dowels in with the same epoxy you used for the original gluing. This will adhere well to both the wood and the existing glue as long as the former has not set too completely.
Variations:
You can clamp the wood together and drill before gluing. Just be aware your clamp needs to be stout for this, as the drill will try to move the relative positions of the pieces. I don't recommend doing this.
You don't have to use wood dowels. I have used brass rod. Especially if the dowel has to be small in diameter, 1/8 inch brass rod from the hobby shop or the hardware works. Just realize epoxy doesn't favor copper alloys much, so you want to chuck the rod in your drill and roughen it a bit with a file or coarse sandpaper before gluing. Steel will adhere better and is stronger, but you don't want to risk rust starting and discoloring your wood.
Epoxies don't give you their best strength in a thin film. In wood, the fact the epoxy penetrates the grain will generally avoid the thin film problem, but I have seen some wood grain soak it too thin. Epoxies designed for wood, like Acraglass, will be the right viscosity to avoid that issue.
In a super-strength situation, I have used System Three Endrot, a super-thin penetrating epoxy that gets deeper into wood grains than the glues, and then glued with that same company's T-88 product before the Endrot had done more than start to thicken. This provides a glued area stronger than the original wood. Just don't paint Endrot too close to the edge of finished wood, as it can creep up under the edge of the old finish and darken it.