There is one method to tighten loose barrels here that was used here often on cheap guns from the 1900 up to the end of WW 2, I did find a number of farm guns repaired using this method though asking friends today none have an example for me to photograph. The method was simple a dovetail slot was filed in the bottom of the barrel hook a piece of steel with the corresponding dovetail shape was slid in and hammered from each side to make the piece a permanent fit in the hook. Next the added piece was filed to the rough hook shape then slowly adjusted until the gun would close on face, I found this repair on a number of abused farm guns well into the 1970's and the guns still locked up tight enough though every other part was well past their use by date so the guns where scraped.
my preferred method is to shim the hook and if done correctly will last many many years. Joe wood suggested solder and the solder method I have used successfully for some fifty the last one being some three years back. I used quality steel shim stock failing that Automobile engine feeler gauges. I see people have used pieces of Aluminium or Steel cans both being extremely unsatisfactory Aluminium flows under pressure also it is soft enough to allow grit to be embedded in its surface making a permanent abrasive barrel hook to ware the knuckle hinge pin. can steel is also soft and will not keep up with the joint ware for a reasonable time.
My method is to first clean the barrel hook and hinge pin then cut a selection of shims, then add each in turn to tighten the action but try to keep it to no more than two. Now the solder I use is 60% Lead 40%Tin Plumbers type not the Electrical grade being 60%Tin 40% Lead being softer also solder paste I have found today available in 60%tin 40% Lead though on your side of the pond things may be different For good solder joints to Steel an "Active Flux" is by far the best active just meaning that it is acidic and must be washed of with water having a little soda added. I just make my own acidic flux by adding a few drops of Hydrochloric Acid to a small amount of the Rosin Flux keep this flux away from blued surfaces. To start I flux the barrel hook ware face then tin the whole surface, unless you are a miracle worker the solder always pools at the bottom of the hook so I wipe the hook with a dry cloth to remove as much solder as possible but if there is still a little left the final step will remove the excess solder. Now for the shim or shim if you need to use two shims allays place the thinnest shin to the barrel hooks surface with the thicker one on top, I do not recommend that you use a flame to tin the shims a soldering iron is far better and it wont over heat the thin shims then wash every thing well.
The simplest way to fit the shims of shim is a follows, place the shims across the hooks mouth and select a metal drill bit that is a good fit in the hook and will also push and hold the shin or shims in place. When you have selected the correct drill place the fluted end in a vice with the smooth chuck end sticking out of the vice to make a hanger for the barrel hook. Apply some flux to the shim and barrel hook mating surfaces (the use of active flux is now not necessary fit he shim shims in tho the hook and then hang on the drill, doing this will push and keep the shims in place in the hook. Heat gently until the solder runs the drill and the weight of the barrel will cause any excess solder in the joint to flow out of the joint then allow to cool, because of the harder type of shim stock I use i find a small diamond file is best to trim any excess shim. As the quality of engineers shim stock and feeler gauges this repair may out live tour good self, I am still using a gin with this type of hook repair I did some thirty years ago and the gun still locks up like a bank vault door. Give it a try though I will not be around in fifteen years for you to say if the repair lasted.
