A bad battery won't blow a starter fuse. The fuse is there to protect the wire, they "blow" by overheating themselves and melting, and a blown fuse means only one thing: overcurrent. That indicates a short of some kind in the device downstream of the fuse, and up-sizing the fuse just lets that condition happen. It is extremely rare for increasing the size of a fuse to be the right way to fix an electrical problem.
There are lots of ways you can get shorts in starters, in both the solenoid and the motor itself. You can even get partial shorts, like partially shorted windings. Those allow the motor to spin but not with enough torque to start the engine.
Don't use this starter until you know what's wrong with it. Up-sizing the fuse is almost always the wrong idea. There is a condition in the aircraft I used to work with where one failure mode of a starter is for it to get stuck "on", and because this component isn't meant to run longer than 15s at a time, the high current starts a fire. To address this failure mode, many aircraft have a "starter contactor" IN ADDITION TO the solenoid itself that the switch wires to. This allows the pilot to force the system off in the event a solenoid short keeps the starter engaged.
Most land vehicles don't have these because unlike in an aircraft, you can pull over to the side of the road... but that doesn't mean they're safer or it happens less. Plenty of engine fires are started by faulty starters. Just Google around and you'll find tons of cases of this happening. Here's just the first one I saw:
Starter causes fire | IH8MUD Forum
"No click" is usually the solenoid portion of the starter. It used to be the standard "fix" was to whack it with a wrench or hammer before turning the key. Nobody climbs under their cars anymore.
Take the starter out and bring it to a shop that specializes in rebuilding them. If in the unlikely event the starter is working, their evaluation will determine that. You can still pay them to clean it, replace brushes, etc, and extend its life (while it's on the bench anyway) and then you can evaluate other components of the starting system (like the cabling). And if it's faulty, then you'll know 100% for sure what to do next - rebuild or replace it.
New starters are cheap on eBay - you can get them for $50-$200, although you have to watch for counterfeits. IMO, a starter is one of the spares people should carry. They're a mechanical item that's prone to failure, it ruins your day when they break, they're not very large, they're (relatively) easy to install (at least compared to something like piston rings), and they're not very expensive.
While it's out, inspect its gear teeth for signs of wear. Broken or missing teeth are a very bad sign that your flywheel ring gear is also damaged. A new starter should not be installed in that case until the flywheel is replaced. You can also have a helper spin the engine with a breaker bar or rotating a wheel while you inspect the flywheel teeth. While the starter is out is the best time to inspect them. It's a pain to do, but better than not knowing.
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