You could do the transplant during the dormant season. This is when professional growers dig them up for shipping so I'm thinking it might be a good time to minimize transplant shock. Although since you're not shipping them in boxes and you'll be careful to keep the roots and soil as intact as possible, I'd bet you could do it in Fall as well. This is much less damaging than what growers do to ship crowns across the country. And by doing it in fall, the plants have some time to reestablish themselves a bit while the weather is cool. Just don't do it when hot weather is around. In summary, I'd say in cool Fall weather or Winter.

Oh yeah... about the seeds, I've never heard of anyone eating asparagus seeds. Mostly people want to avoid them since they only serve to draw energy away from spear production and because they produce weeds.