Hopefully nobody has applied heat to that lever as that wouldn't be good! How difficult is it to slip the snug one back off? That would tell you if it's the (one) new shaft or a damaged lever.
I would be inclined to determine exactly what the problem is and correct it. The sloppier that fit is, the more work that small pin/rivet is doing. Having that rivet shear under hard braking would not be good!
As a last resort you can drill a very small hole ALONG the shaft at the top, in the seam between the shaft and lever so the hole is half in the lever and half in the shaft. Then tap in a slightly larger (press fit) pin. Done properly this will both tighten the fit so there is no side motion to the lever and the pin will act somewhat as a key helping to prevent any rotation.
The drilling and the pin length should be no longer than the width of the lever at that point so the pin does not protrude past the base of the lever and into or touching the bushing in the backing plate. The pin really should not need to be more than about 1/6" diameter. The end being inserted must have a slight chamfer.