The wire harness in the X1/9 doesn't handle heavy current draw very well, especially when you have the heater fan on high, rear window defogger, wipers, dash lights, headlights, turn signal, and heaven's don't hit the brake pedal!!!
This is pretty normal.
My '78 X1/9 has less trouble, different harness, different power distribution than the later cars. My '82 X1/9 has the EXACT problem yours does, and, the simple way to fix is is isolate this supply and power it directly (with a fuse) from the battery. I have done this with my headlights, as I have 80/100 watt H4 lamps in my car, and the original harness gets toasty with that much current going through it.
My alternator on my '78 doesn't put out as much charge when it gets wet, yes, something is shorting, but it is minor. It drops to 11 volts or so but the battery can keep the car running for a very long time even with the headlights on.
On my race car, I have a high output alternator to keep the gel cell charged up and keep the ignition going. I used a A/C Fiat Spider alternator, and modified it to fit into the X1/9 casting. Apparently the A/C one has more available current, works wonders and hasn't failed yet.