The larger cage gives them all more room so that they are not literally fighting for space. Hormones to induce estrous was also recommended as then you are almost assured of a plug after one night so that the females can be removed from the cage as soon as possible. Placing the male and female in different parts of a divided cage so that he adjusts to her scent before she is placed in the same cage with him. Mild sedation was suggested, eg valium. But I think this would be a little difficult to get past the animal control groups in some areas.
More extreme measures involve removing one of the testes of the male for in vitro fertilization.
This suggestion came with the offer of the protocols if needed but as this person did not reply to the list I am wary of giving her e-mail address. Again, thanks to everyone who replied.