I suspect your converter is a WFCO converter, rather than a WAFO. If you want to throw money at it, there are better converters available, but I haven't seen anything you've written that suggests it's not working properly nor adequate for typical uses.
Wiring loss of 1.5V is not normal. If you have 13.7V at the converter output but only 12.2V at the batteries at the same time, there is a poor (high-resistance) or open connection between the two somewhere. The two places should be at about the same voltage, certainly within a few tenths of a volt.
The experiment with running the engine for awhile to charge the batteries shows that they were indeed being charged from the alternator. 20 minutes is not enough time to charge them fully, or even close to fully, if they're anything more than slightly discharged, so there's not much to guess at their condition based on that. I don't think they were behaving as though there was a shorted cell or other similar problem that would make them completely useless, but they may well be not in great shape still. Regardless, even if they are worn, that is not your only problem.
I would trace the wiring between the converter and the battery and measure voltages to find where the voltage drop is happening, and fix whatever is wrong when found. It's not always a the easiest task as the wires are rarely easily accessible, but I don't know of a better way to find the problem. It's almost certainly at some connection between parts or wires, rather than in the middle of a wire, unless possibly there's some obvious damage to a wire.