You don't have to add more sodium bromide after first adding it after a water change, but that is not the same as adding an oxidizer to create more bromine. Bromine gets used up even when you are not using the spa, but it reverts back to bromide ion. So you need to re-activate it using an oxidizer to convert bromide back to bromine.
So I repeat -- you add the sodium bromide once after a refill. You add an oxidizer regularly after every soak and as needed in between soaks.
As for your adding 1.25 cups of what I presume to be 8.25% bleach to a 300 gallon tub, that would be 27 ppm FC in a chlorine test or 61 ppm Total Bromine in a bromine test (the units of measurement are different for chlorine and bromine). I suspect that this very high amount is bleaching out your DPD indicator dye. Did you see a "flash of pink" when you added the DPD powder? If so, add more DPD powder (another scoop or two) to see if you get a color. If it's truly zero, then you have something in your tub consuming a lot of bromine. Also note that chlorine and bromine tests cannot distinguish between chlorine and bromine. So even if you did not have enough bromide in the water to make bromine, the chlorine you added should still have registered on the test.