Have you checked the jets? Worth giving them a check and clean.
Also, why don't you just run a temporary fuel line straight from the tank pick up, via a new pump, straight into the carbs; that way you can eliminate (almost) everything from the tank to the carbs and work out which end the problem is coming from.
I had a problem with my Facet Red Top lift pump last year (tank to swirl pot). Also, remember they have an internal filter in them, worth checking. My pump, after some decent checking had had its day, so bought a new one and problems solved. It used to work for a while and then just crap out, then restart.
I think that the cause of the lift pump failing was perhaps the crap in the tank, it could also been from the affects of blocking the thin gauze that sits around the pick up inside the tank. What I ended up doing with mine, is neatly cutting the seat floor above where the tank pick up is, so I don't have to drop the tank if I want to get into / service anything in the petrol tank. The hole in the back seat area is about 6" diameter and I was concerned that it might be an MOT issue - but clearly not.
I can't send you a photo of this - as the car is in the garage as you know.