Theory B, and rather simpler:
The duff relay powered both something on which the immobiliser depends and the instruments. (And would probably be the mysterious "Ignition relay" on Sigmas.)
How to tell these theories apart:
• Work out which relay does what.
• Rerun the voltage test. If it's still dropping to 10.7 V on cranking but now starting nicely then the first theory is wrong. (I'd also recommend routinely measuring battery voltage at rest, minimum during cranking, and at 3,000 rpm in the same way as fluid levels and tyre pressures... )
Jonathan