So, when it starts to stumble you could try pulling the choke to see if it smooths out any. That might tell you if it's going lean due to fuel issues.
I tend to think it more likely an electrical issue.
Unless it has been converted to electronic (like Pertronix or similar) I suspect the condenser may be the problem - If it's been converted to electronic, there is no condenser. It also could be the coil. Both of these items can fail when hot and work OK again when they cool off. Easiest way to test is to simply swap out new or known good parts. Most of us travel with spare condensers, coils, rotors, caps for this very reason.
Wherever you're located, you may want to join the nearest Healey club to have some helpful resources nearby.