Just a thought that might be worth trying.

Stick a lot more fuel in it before spending too much on faults. Had similar years ago on the vauxhalls, if you ran them out of fuel they'd not run right if you just stuck a can in, even though you'd have been ok running them about with that much in. Most tanks have a baffled area where the sock on the pump sits and you need to get a decent amount in.

Might just be that as you go forward e fuel is moving away from the pump (if it's at the front) and if you go backwards the fuel feeds the pump easier.

Might be worht a shot. We used to have to a third of a tank back in the old vauxhalls if they ran out.