First off, use toilet paper and pack the cleaning kit space in the butt stock. Now figure out how much shot you need to add to even the rifle out. The toilet paper will keep the lead all the way back against the trap door (were you want the weight to be). Once you figured out how much shot is needed, remove the butt stock from the rifle and pull the butt plate.
Now grease the hell out of the cleaning cavity plastic (read release agent), and mix up enough epoxy that when the shot is added, all of it is wet. Now using TP again, pack the bottom of the cavity in the stock that when the epoxy'd shot is added it will come to the top of the cleaning kit cavity. If you have a void between the top or the shot and the top of the cleaning kit space, add more epoxy on top of the shot. As the epoxy starts to set up, you can run a knife across the top to remove the hump (capillary adhesion effect).
After the epoxy start to set up, give the butt stock a good tap to get the shaped weight out, then let the weight sit for a day or two (the epoxy will still be hot, and needs to cool/fully cure. At this point, you can clean out the grease from the cleaning kit space.
Then all you need to do is re-install the weight when you put the rifle back together.
On the back of the weight/in front of the trap door, use a piece of weather stripping sticky back foam to keep the wedge shaped weight from bounding back against the trap door. Also, if you plan on being able to remove the weight from time to time, you will need to sand down the top edges of the weight to allow it to fit out the trap door.