So this is what I understood from your discription:
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Assuming your inverter is the only load, the fuse at the battery should match the max current the inverter manual says. This fuse should be as close to the battery as possible. The wire size between the battery and the inverter must be at least large enough to handle the current the fuse allows (preferably a bit larger). If the run is greater than a few feet (round trip from the battery to inverter and back), the wire size will be larger so you will need to look up the wire size for your current and distance. (
This site has a nice chart you can use to look up wire size for various round trip lengths. You should use the critical voltage - 3% drop values from the chart)
The fuse for the charge controller should be slightly larger than the max current expected from the controller. This fuse should be placed as close to where it branches from the other circuit as possible. The wires from the charge controller to the battery must be at least large enough to handle the current the fuse will allow. If the run is greater than a few feet (round trip from the battery to controller and back), you will need to look up the larger wire size for your current and distance.
With just two panels in parallel, I am not sure a fuse is needed (or even feasable) on the panel circuit. However, the wires must be large enough to handle the full current from both panels. Even though you may not need a fuse, a lot of people like to put in a disconnect.
