Small Engines (Lawn Mowers, etc.): Briggs 17hp OHV backfiring (model 313707), piston rings, leak down test

Question
Took mower to 2 shops, could not find problem. Mower will start but runs rough. If you lower the engine speed,motor will run smooth and will continue to run smooth as you increase speed. When blade is engaged,motor shuts down. When restarted,motor sputters and continually backfires,shakes whole mower.Tried new carb,spark plug and coil. Checked for bent push rods-found to be ok.Adjusted valves and timing.
Recommending I trash this 5 yr old Murray. Any ideas?

Answer
Sorry it's taken so long but their website has not been working all weekend.

Have any of the shops performed a leak down test?  This will check the valves, piston, rings.
Have you checked the flywheel key?
Can you send me the type and code number as well.

Eric