In my former life I tested automotive brakes, so I got thinking about the relining problem. I have reservations about using the flexible lining with rivets. You might consider this. JB Weld epoxy is available in automotive stores. It can withstand high temperatures from engine blocks and will hold torque when used to rethread a stripped hole.
Clean the surface of the brake shoes. Install the brake shoes on the brake backing plate. Slip a piece of wax paper between the shoes and the backing plate to keep the shoes from being bonded to the backing plate. Obtain two good hose clamps sufficient in diameter to surround the brake shoes. Cut the brake lining to the required length, apply a minimal amount of epoxy to the surface of the brake lining (enough to squeeze out a bit , but not too much). Slip a hose clamp around the brake shoes with enough room to slip in the brake linings. Position the brake linings and tighten the clamp. Check the position of the linings and re-adjust if necessary. After the first clamp has been tightened, install the second clamp to give full pressure on the complete surface of the brake linings. Wait 24 hours, remove clamps and clean off excess epoxy and trim the brake linings at the edges if necessary.
If the diameter of the linings is too large to install, use a strip of emery cloth or sandpaper in the old shoe shining method and reduce the thickness of the linings equally on each shoe until the desired fit is accomplished.
There is probably no way to determine the longevity of the lining, so it is advisable to check the shoes on a regular basis.