I think that it's worthwhile to try using only bleach for a while to see if it works for you. Bromine does not lose as much to sunlight as unstabilized chlorine because the higher pKa means that there's less hypobromite than hypochlorite at the same pH. It's the hypochlorite or hypobromite that is primarily lost to sunlight. Continuous dosing would work better than periodic dosing. That way you can maintain a lower but steady bromine level. A peristaltic pump would be a good choice.
You could try using a salt system as that would give you a steady dose of chlorine/bromine. I have seen that work. The bromine level usually does not get very high, but the continuous gain and loss reaches an equilibrium. The cell will not last as long due to needing to run it at a higher percentage.
Note that most manufacturers don't recommend using their salt systems on bromine pools. Therefore, you're going outside their requirements and probably void the warranty if it should be necessary to make a claim and they know you're using it for a non-standard purpose.
Also, I have seen where the addition of cyanuric acid seemed to help maintain bromine levels. Although cyanuric acid does not bind to bromine, it might protect bromine at lower depths by making the water more opaque to UV. However, that's not a known fact.
Another complicating factor is the presence of dmh, which is the carrier chemical for the bromine tabs. Some bromine probably binds to dmh in the same way that chlorine binds to cyanuric acid. It might provide some sun protection and it probably reduces the effectiveness of the bromine. Fist, we don't know how much dmh is in the water. Although, we could estimate the amount from the total weight of tabs used. Second, we don't know the equilibrium constants for bromine-dmh. And third, we don't know if bromine bound to dmh retains any significant oxidizing or sanitizing capacity.
Long term, it would be better to drain and refill to get on a stabilized chlorine regimen.