I have had similar problems with the electronic ignition on
my 'B and it turned out to be the wiring for the inductive
pickup coil inside the distributor. After years of flexing when the
contact plate rotated with the vacuum advance, one of the
wires which runs from the pickup coil to the rubber plug in the
side of the distributor, broke. I patched the first time about
two years ago. Last fall, I had a similar problem and discovered
that there was only a single strand of wire making a connection.
BTW, my 'B has the later style electronic ignition with the
amplifier mounted on the inside of the right side wheel well.
Hope this helps.
1963 Austin-Healey 3000 Mk II