Jim,
The S1's were notorious for their bad electrics but especially for the bad earthing.
There are about 7 earthing locations around the car. Some locations have multiple earth points. I've never had an S1 so these locations are based on my knowledge of Series 2 cars:
There are multiple earthing points on the front inner wings both offside and nearside. You'll find them behind the headlights and forward of the LHM tank.
There are earthing points on the rear inner wings accessible from the hatches used to access the lighting clusters.
There's an earthing point inside the boot lid and up behind the interior light.
For each of them you need to remove them first, clean them up with a combination of wire brush and emery cloth and grease them well (even better if with an electrically conductive grease).
Although this is not guaranteed to solve your problem its your best starting point given the model's electrical history.
Whilst you're at it there are two reasonably thick cables jointed at the -ve battery terminal together with the very thick one which goes to the engine for the starter motor. Those two cables are earthed on the body just forward of the LHM tank. They are prone to corrosion.
Hope this helps. If you clean up all these connections and you still have a problem just let us know.
Cheers
noz