I've considered doing this and actually have Ahh some on hand, but have not used it yet. My first shot was going to be to run ahh some, drain, and then spray water in each jet hole and such and drain any residual water. Use bromine again, to make sure it was a biofilm issue or something (we bought the tub used but put a new filter in).
If I am still having issues after that, I figured I'd swap to chlorine. I just like the convenience of the spa frog inline system with the bromine cartridge and mineral cartridge. I know that now make chlorine cartridges, but I reuse the bromine by opening them up and putting bromine tabs in them. I don't think there is a product out there that I can do the same thing with chlorine tabs because most dissolve to fast.
It sounds like you drained the tub three times, that seems a little excessive honestly. I was only going to do it once, maybe twice max. That's a heck of a lot of water!