first of all I see this is your first post so welcome to the forum from mid California.
I suggest you start with not allowing your battery to go so low taking it to 10.5 will ruin the battery and seriously shorten it's usable life time. You should not let it get get below 50% charge which is (believe it or not) about 12 volts, if the fuse blowing is related to the low battery when starting the gen this may solve your problem.
Try turning off all the systems the battery and gen feed before starting the gen (when you have a very low battery). I am thinking the sudden rush of current could be what is causing the fuse to blow.
Dale & Terri, Lulu & Tiki (our Chihuahua's), New rescue puppy Prince - Pom/Pug mix.
2013 Tundra, Double Cab, 5.7, TRD Off-Road, 4X4, Full Tow Package
2013 Wildwood T26TBSS - Sold
2000 Gulf Stream Sun Voyager M-8357 MH, Ford Trident V10 Gasser, 35 foot.