You say that it's 'popping' which could be a timing issue. It's quite common for the flywheel key to partly shear, knocking the timing out slightly and can result in 'popping' and backfiring!
Another issue on some B&S engines is the auto decompression unit in the crankcase. It makes for easier starting but they wear out and can lead to popping. We have a ride-on with a 13hp B&S engine that was 'popping' when you changes the revs. I did most of the things you have already done to your engine & the only thing that helped was increasing the valve clearances. It still pops occasionally but I think that's down to the auto decompression unit being worn & just live with it.
Another issue on some B&S engines is the auto decompression unit in the crankcase. It makes for easier starting but they wear out and can lead to popping. We have a ride-on with a 13hp B&S engine that was 'popping' when you changes the revs. I did most of the things you have already done to your engine & the only thing that helped was increasing the valve clearances. It still pops occasionally but I think that's down to the auto decompression unit being worn & just live with it.