Put the bit in an indexer and use a carbide mill to make the shank triangular. That won't slip in the chuck.
Proper reduced shank drills already arrive with three flats.
Pushed too hard, they don't "slip" in the chuck.
They simply ruin it.
Smooth shanks at least preserve the chuck.
Be patient. Take "baby bites", constantly clear chip and re-lube, and you can get away with a lot.
The proper replacement is a chuck that is built for that service.
Typically cannot grasp a drill under ~ 3/16"// 13/64,
..but CAN grasp 3/4".
If a Llambrich Hybrid SPH20J3 is not in your budget?
San Ou "Harvest" makes a half-decent "workalike" as "3/16-3/4 JT3".