I don’t know if you have ever seen those ‘Rapid Fire’ products that used to be sold about 15 or 20 years ago? These were clamped to the trigger guard and had a lever which ended in a ring. You would grip the gun as usual but put your trigger finger in the ring and pull the lever. The lever was connected to a cog which turned and activated the gun’s trigger a number of times for every pull on the lever.
The BATF ruled these were legal as they were not part of the gun and the gun’s trigger did not discharge the gun more than one time per activation of the trigger.
If I were so interested in having a burst-fire handgun I would look into one of these. I would also do the paperwork and pay the $200 tax to convert the handgun into a Short Barrel Rifle. That way you could legally add both a buttstock and forward grip to the handgun and give some measure of controllability to the package in burst fire.