look at this picture , what do you see ? a row of stitches not 2mm from the bottom of the strap!!! you should have sewn the bottom there, with maybe a second row and sewn the top and been done with it.
This isn't rocket science friend leather straps have been sewn on for thousands of years. any one of the suggestions will work they are from tried and true leather workers with years of experience.
rivets will work in leather for some things but not all things and that is determined by the direction of the load on the rivet. the load has to be on the shank not on the heads if that makes sense. such as two pieces lapped together on a belt pulling from each end as compared to your strap which has the load pulling out against the heads.
Using them as Fred and AL suggests changes that load factor, the load is attempting to pull the strap then down through the slot not out against the head of the rivet.