You sure this isn't electrical?

No fuel filter will have deposited all manner of layers of shite in the carb. No wonder the hand trick didn't work

Backfires are from unburnt fuel in the exhaust, hence f you have blocked jets excess fuel in the exhaust is unlikely,sounds more like electrical as Russell said.Maybe condenser breaking down,loose wire on coil or something else breaking the power in the ignition circuit. Sounds like a case of process of elimination..

If it doesn't pull when foot flat down - it will highly likely be crap in the jets - I fanny'd around for ages to get the Dyane to run well, changing the ignition, timing, tappets, etc and it always ended up with taking the jets out and cleaning the crap out of them, even if microscopic, and the bowls...still got crap in them despite the filter in the tank and an inline one about 150mm from the inlet on the carb.
Although......a thought, you haven't got a hole anywhere in the inlet manifolds or a leak from the gaskets under the carb or where they go into the heads?

I'm beginning to think electrical too. It's a Solex 40 so it's fairly basic, although I'm not convinced I found the secondary jet.

Could be the coil or the condensor, ill start with the coil then change the points and condensor and re-time her.

To be clear, she's not backfiring, but popping in the carb, not the exhaust.

if there is only one big hole then there isnt a secondary jet

primary and secondary refer to the 2 venturis and the associated jets in a twin choke
"progressive" carb so blocked 2nd jet not likley the problem this time

Popping carb a sign of advanced timing or perhaps a weak mixture and definitley look at the tappets.

Ah ah......not a twin choke then...could still be crud floating around but I'm thinking timing may have slipped.....

I did wonder. The primary and only jet I thought was clear, but when I stuck a light in it you can see it dribbling rather than squirting. It squirts when you rev up, then just dribbles pathetically.

I shall however look at timing and tappets.

compression test?

IT, not HER FFS.