Use chlorine tablets. You can shock it and run the pump with all jets and air inlets (or is it outlets) open. There is no reason you cannot use chlorine instead of bromine. Back when we had one we used both at various times. The tablets made for hot tubs are 1" vs the 3" for swimming pools. It is a LOT easier to keep the chemicals in a pool properly balanced as the water volume is so much higher. With the much smaller size of the hot tub, a bit too muc of a chemical can really throw it off balance.
As my first post said, you really should take a sample of the water to the pool store and have them do a thorough analysis. Be sure to explain your problem. You may need to use an algecide (I know that algae tends to build up in some of the crevasses of our pool if we are not thorough. There are algecides made with silver and others with copper. Offhand can't recall which one I prefer, but it gets the stubborn algae.