Maybe I missed it (or didn't understand it clearly...your 3rd paragraph was a bit odd), but you didn't say whether it works properly
on JUST battery power alone (e.g. even though it says 20 hours remaining, how long does it actually
run on battery power? And, with the AC adapter unplugged, will the laptop boot-up and run?)
If it doesn't power-up on AC power even when the last pre-power-off indication was that the battery is "fully charged", then yes, the first thing I'd do is BORROW a different known good battery and try that.
If it works properly, then I'd verify everything else regarding power usage is working properly in various power modes: runs off battery only, properly charges battery, battery indicator shows more realistic values, indicator shows battery level actually decreases while in use, etc.
I wouldn't buy an overpriced battery just to try the experiment.
added clarifying sentence
edit to add:
re: running without a battery installed...
I've never run across one of these, but I've read some laptops apparently do need the battery to have at least "some" level of charge in order to operate properly. Maybe that's a way to make sure customers keep buying batteries instead of basically turning their laptops into desktop computers