Generally speaking, situations like that just cause inflation. Items start to be worth more money because the money is so common, and there’s far more cash spread between far fewer people.
It makes things more difficult since you need to carry around massive wads of cash to buy things from other survivors, but that’s not too big of a deal. Still better than minted silver or something, and it means that merchants setting up a compound and a place to store their own (and others’) money would be quite feasible.
Getting a massive pile of cash would still be fairly difficult. It would still require breaking open a bank vault or something, it’s not like the apocalypse undid all security measures humans go to to protect their money.
It’s pretty much a given that once some semblance of trading begins, some form of currency would be required, and like you’ve said, whatever the traders use is what everyone will use. Ammunition would be one option, but then you have the issue of literally throwing money at things that attack you, trying to balance between defence and cost-effectiveness, things like that.
Plus, if your currency is something practical, then it only has value to those that need to do that thing, so you run the risk of your currency losing value once everyone, for example, has solar powered laser guns. Also, anything you use is going to be excessively bulky and heavy if you manage to amass a decent amount of value. It doesn’t actually make any more sense than just bartering.
Bottle caps are still chunks of metal, so they’re bulky and heavy as well. In their case they are still only valuable as currency, and they’re inferior to actual money.
Current money is specifically designed to be good currency. It’s plentiful, but limited, and pretty much impossible to counterfeit post-apocalypse. Honestly, I’m not sure why Fallout uses the bottlecap system.
Tl:Dr Money is actually pretty good money. It would make sense to keep using it as money.