I guess that *would* work nicely, but I already removed the rectifier and did the mods I posted this morning, and that didn't go as sweetly as imagined... It was welded in with that awful lead-free solder, and thus was quite a messy removal, only one leg survived.
The PCB location of the minus pin (-) on the rectifier is already tied to ground, so the 1000µF cap is still working as a filter.
I guess I could buy another rectifier if it doesn't work as expected, I may even have one kicking around in the drawer...