Most car batteries these days are of the SLA (SEALED Lead Acid) class, so there's no checking things like fluids these days; it's dead once it stops working right. If it's charging fine on the external charger, it could just be a tired battery that fails under a load but works when off load. It's not a common failure mode but it happens.
The first thing I would do is check the terminals for battery acid and if you see any, clean it off; same for the battery. If it helps, sometimes the buildup can also cause this but generally, once they start outgassing it can sometimes mean the battery is just flat out on the way out. Put dielectric grease on both parts to avoid further issues, as well as any new batteries you buy moving forward.
If that doesn't help, get the codes on the car scanned and check for issues like a bad alternator; the alternator charges the battery. If the alternator looks good, it could be a wiring or ECU issue. The big thing to remember with the DMM test is to use a good meter; the alternator can blow out a cheap meter really quickly, especially barely qualifying CatII ones.