I bleach selectively a lot, and occasionally bleach-back an entire print. I use a potassium ferricyanide/potassium bromide rehalogenating bleach. The technique is definitely usable and, especially with selective bleaching, a standard tool in many fine printer's toolbox.
As Ian points out, you may want to pull the print just before it reaches the stage that pleases you the most. Give it a thorough rinse in running water (5 min) and then re-fix.
And, yes, you do need to refix after bleaching. The bleaching will leave unfixed halides in the paper that will yellow with time, just like an underfixed print.
Best,
Doremus