I would agree that the battery isn't charging (or won't hold a charge), the Tenere ignition is CDI and doesn't use the battery.
I wouldn't assume that the alternator is at fault though, the bike has old wiring, and I would give that a good look over first.
You can measure the resistance of the alternator windings with a multimeter and check they match the figure in the Clymer manual, sorry I don’t have it to hand.
Also try charging the battery off the bike then check it with a battery hydrometer, a small one designed for bikes. Check the electrolyte level first , if its very low the battery is probably scrap
The other thing in the chain is the rectifier/regulator (or is it 2 separate units, can’t remember that either), I’ve had one fail, if ignored they can destroy the alternator windings too, so defiantly worth “a stitch in time”
I’m assuming that you aren’t running a couple of spot lights and a heated waist coat, though I had no problem with heated grips