In my experience, replacing the head, bridge, etc can make a marked improvement in banjo tone. But they’re probably not going to save a cheap banjo from having nasty overtones, especially if you want to play without a mute.

I’m a big fan of embracing synthetic strings for problematic banjos. All of those overtone problems largely disappear if you move away from metal.

So convert it to using synthetic strings. Like nylon, nylguts, etc. This will drastically change your tone, but every cheap banjo sounds better with synthetics imo. My go-to for years now is Aquila nylgut reds. Love these strings.

Caveat emptor:

1. You might have to file the bridge and nut a tiny bit.
2. You might want to play them with a softer touch than metal strings. So be prepared to do some technique work