Larry,

An Isolator allows current to flow from the center post (Alternator) to the outer posts (Chassis and House batteries). It blocks current from flowing from the battery terminals to the alternator or to each other. Therefore the house loads cannot discharge the chassis battery, but the alternator can charge both batteries.

Since you see ~15 volts at the center post (alternator) but less than 12 volts on the chassis side, that indicates to me that the Isolator is faulty. The battery terminals should be about 0.7 volts lower than the alternator terminal with the engine running. That's the only test you really need to indicate a good/bad isolator.

Isolators are widely available, I'd suggest Princess Automotive or Canadian Tire for us Canadians. Get one rated at at least 90 Amps, 120 or 150 amp models give you some more robustness.

The center terminal is generally the Alternator. It doesn't matter which battery connects to the other two terminals.