Fuse 35 watt system?

Welcome to the forum ZaptheCat!

First, the battery is the source of high current in most off grid systems... So, you should have a fuse/circuit breaker based on the wiring leaving the + battery bus... If you have a 1 amp load, you should have at a minimum 1.25x 1 amp = 1.25 fuse rounded up to 2 amps -- With heavy enough wire to both carry 2 amps, and not have too much voltage drop from battery to the lights.

If you use a voltage drop calculator, say 1 amp, less than 1/2 volt drop maximum and send the power 20':

http://www.calculator.net/voltage-drop-calculator.html

18 AWG wire:
Voltage drop: 0.26
Voltage drop percentage: 2.17%
Voltage at the end: 11.74

For the charge controller:

10 x 0.19 amps * 1.25 NEC wiring derating * 1.25 solar derating = 3.0 amps minimum branch circuit rating (fuse/wiring).

Keep wire run from charge controller to battery bank relatively short... Ideally, you want around 0.05 to 0.10 volt maximum drop from controller to battery bank (want charge controller to accurately measure battery bank voltage). Voltage drop calculator--Using 14 AWG wire, 3 amps, 4 feet:

Voltage drop: 0.061
Voltage drop percentage: 0.51%
Voltage at the end: 11.939

Of course, with a 14 AWG wire, you could go as high as 15 amp fuse/breaker.

Lastly, there is fusing of the solar array itself... For larger panels (100's of watts), we generally have to fuse/breaker each panel if there are three or more panels in parallel... One shorted panel is fed by the rest of the good panels--Need a fuse/breaker to protect the wiring from over heating the shorted panel.

With your very small solar panels, I am not sure what the series fuse rating for those panels is--You probably would not worry about fusing these small panels...

-Bill