I'm really wondering if take the pump out of the system.
And ... (did you overlook finishing the sentence?).
Beer in bag and have it above the beer engine then apply pressure to the bag with some weight. Brick books or something else, the switch you mentioned when triggered will allow beer to flow through the beer engine.
I suspect the pressure generated by the pump is taking the place of the cylinder draw pressure and you would need to make this pressure by external means.
I have in mind "weighting" the beer in the bag, but only to keep some CO
2 condition in the beer: The electric (or air driven) pump will provide propulsion in place of the hand-pump's no-existent cylinder mechanism. Some resistance in the serving line, such as an up-stream check-valve (a real one, not a hand-pump demand-valve) and a "sparkler" on the nozzle should produce the turbulence (and CO
2 breakout) to emulate a hand-pump cylinder. The latter might need some tweaking to get right, but that will be easy undemanding tweaks.
I'd been considering pumps following my "Stitch Ale" experiment last year. And then along comes this "sham" hand-pump query with a pump associated, and I find my objection to "sham" hand-pumps vanishes. "Sham" hand-pumps
don't have to serve fizzy beer (as I had thought), with an electric pump they can serve low-carbonated ("cask conditioned") beer as if out of a hand-pump; but without the significant workload of managing a "real" hand-pump in a home-brew situation. Wish I'd thought of it before.
@Three Guinea Pigs Pub: Firstly, have you changed your username? If not, apologies for previously addressing you as "The Guinea Pig Pub"! Earlier you said the pump you have is not self-priming? Don't worry, they are! It's a "diaphragm pump", and if it wasn't self-priming it wouldn't pump either (it's how they work). You've picked up some misleading info there.
I may suggest you use an "impeller" pump instead (the common, often magnetically driven, type) because of the turbulence they create, but a diaphragm pump has the advantage of being self-priming and "seals behind itself" so works as a "check-valve" too. Six of one, half dozen of the other, at the moment.