So had another bash at fixing my SID unit today.
Took it apart again, but this time I noticed that the actual display screen can be detached from the plastic unit that holds it in place, allowing access to the other end of the circuit ribbon.
There's also a thick rubber band that applies pressure to the ribbon from behind, this was squashed flat, so I moved it round so that fresh rubber was pressing on the ribbon and for good measure added some packing.
Re-assembled, re-installed and 'Hey Presto!' I have a fully functional SID unit
(Took some pictures if anyone is interested).
So I thought I'd have a go at the climate control display. Unfortunately there are two star bit screws that I don't have the right size bit for (must be tiny). So no progress with that as yet until I can find the right size star bit.