Tips for rework.
Voltage regulation - Take the 22R resistor and put in series with the diode to battery, to provide some current limiting. I have added simple voltage regulation using 1W zener diodes, a 3V3 and a 3V6 zener in series ( buy around 10 of each and use a power supply and a 100R from 6v to select pairs of diodes that have 6V9 drop, and place them from the junction of the 22R resistor to ground via a 1N400x diode. this will clamp the max voltage to the battery to 6V9, enough to keep it charged and not too high to keep the battery life long. You do not need extra filtering, a little ripple on a SLA charge is not a worry. You probably would replace the OHL 100uF cap with a bigger size same capacitance unit to have a longer life, use a 25V unit to have better voltage margin.
To the light side, you probably want to add a 1R2-1R5 1W resistor in series with the big panel array, and a 10R 1W in series with the small array. Current limiting to make the LED's last longer, though they are already degraded.
To flash the array you take a standard 555 astable, running at 0.5Hz or so, with a diode to have a short high time and a long low time, and use a Mosfet to do low side switching to the common of the array ( this then allows the selection of what flashes via the switch) or make the 4 LED array flash always when power is lost, the switch then enabling use of the big array or not, small always ready to flash on power loss. When using a 555 Ic you will need to have a 1000uF 16V capacitor across the battery terminals for extra smoothing. This will make the 555 happy.
If you go for the ultracaps you can use the zener limiter, and just do a direct replacement of the battery with the 3 ultracaps in series. You do not need the extra capacitance with the flasher in this case.
Addons would be a mains fuse in the marked place on the PCB, and a 240v varistor across the transformer primary after the fuse.
Edit, looking at power dissipation replace the 22R 3W resistor with a 47R 5W wirewound unit, as this will run cooler, and will limit the zener current to 100mA when the battery is full, or disconnected. Will also limit the current to what the transformer can really supply, not the 200 plus ma you were drawing from a 100mA unit. Leave the leads on the zener diodes long to allow them to dissipate heat, or solder on some copper heatsink tabs near the diodes.