Suggestion for fixing leaky exterior door handle
I would remove the handle, clean it as well as the door, and use an exterior grade silicone caulking (and plenty of it; enough to squeeze out a little) on the inside of the bell's perimeter as well as screw holes of the outside handle, and re-install it. Clean up any squeeze out.

I would not caulk the inside side of the knob as you want it to leak again to alert you that is leaking once more (your fix has failed) rather than your door becoming squishy.