Thought of one - Angel of Scavenging. You cannot reload weapons, so whenever one is empty, you just throw it away and look for a new one.
I'm not sure if overloading should still work. Either way, it means that picking up ammo will be unnecessary. Instead, picking up multiple guns of the same type will be required. There are tonnes of lower level weapons just lying around, so that won't be a problem. There are even kindly former commandos handy to provide multiple plasma rifles. Rocket launchers and BFGs will be much harder to recoup.
Perhaps generate more weapon drops to make things fair?
Difficulty - MEDIUM. If all else fails, that chainsaw will still go on forever.
And on a side-note, customisable challenge modes sound like fun. Long live the promised API!