The converter takes 110v shore power and converts it to 12v to charge the battery and power 12v items like the lights, circuit board in the fridge, furnace, etc.
If the battery you bought in April didn't die until Sept that tells me the converter is working but not putting out as much power as it should. Had the converter not been working at all, (as in a blown fuse) the battery would have died in a week. So I don't think this is a simple fuse issue, the converter is bad and needs to be replaced.
A good choice is the
Progressive Dynamics 4600 series. They are made to replace your stock converter without having to remove the entire electrical panel box. Open the cover to the fuse panel and see what brand you have now, then call the guys at
Best Converter and they will get you the right replacement. These come with step-by-step instructions. All you do is remove the fuse panel cover, disconnect 2 wires, slide out the old converter and slide in the new one. If you are the least bit handy, or have a handy neighbor this is an easy job.