Push-in introduction cages, work best, when there is emerging worker brood inside the cage with the new queen, and perhaps a little nectar and pollen (beebread). The newly emerging workers will immediately adopt the queen, as theirs, and begin taking care of her. It helps if there are cells of food in there also, so they can all get to food without having to rely on the hive bees to provide it for them - they may be too slow to do this.
It will likely work, but sure sounds like a complicated way to do it. In reading your plan, I wasn't sure which you were placing atop, I would recommend placing the suspected laying worker colony above the queenright colony - so brood pheromones can be drawn up as the warm air rises from the healthy brood nest.
Laying workers aren't rare, nearly all colonies have some laying workers present. They only rarely become a problem, when a colony has been broodless for an extended period of time (the presence of the normal worker brood and their associated brood hormones keep the nurse bees on-track, removing laying worker eggs from worker cells).
It will likely work, but sure sounds like a complicated way to do it. In reading your plan, I wasn't sure which you were placing atop, I would recommend placing the suspected laying worker colony above the queenright colony - so brood pheromones can be drawn up as the warm air rises from the healthy brood nest.
Laying workers aren't rare, nearly all colonies have some laying workers present. They only rarely become a problem, when a colony has been broodless for an extended period of time (the presence of the normal worker brood and their associated brood hormones keep the nurse bees on-track, removing laying worker eggs from worker cells).