If you buy a mains charger you won’t need a PSU. Look for something with a bit of power, 80w minimum for larger capacity batteries. There’s plenty under $80 that will do a great job. Get a two output one if possible, it will speed up your recharge time.
Cheap chargers are usually designed only for tiny packs, you won’t need to break the bank completely but you are not going to get a good one for $20.
I have charged batteries away from my mains supply once in 6 years, charging those big packs off a car battery is a good way to end up with a flat car battery! Buy more packs, charge at home, that’s how I have usually done it.