the battery... it has to take a charge and hold the voltage, and act as a buffer when connected to a charger or converter that kicks in and out (over-active), but dont overlook a circuit breaker that may be kicking in and out, so that sometimes the lights are being powered from the battery, and then when the circuit breaker resets, it allows the stronger voltage from the converter into to the light circuit...
or it could be working just opposite of this, so when the lights are bright, the circuit breaker has tripped and the circuit has been disconnected from a ground fault somewhere on th eother side of the breaker... and when the circuit breaker resets, the circuit again goes to ground and dims the lights, until the breaker trips again...
somewhere in here you may fine the answer to why the other fuses are blowing..