Asparagus self-sows readily. Your intentions are admirable, but you'd be better served to just prepare a suitable asparagus bed and plant or sow the plants/seeds in a convenient place than to coddle a plant in an inappropriate place. However, though the root system is often very immense, if you really wish, you could move in the fall (allowing growth till then w/o mowing it).

Typically an asparagus bed takes at least three years to come into full production, but then it will produce more and more each year if properly cared for. It will also persist a very long time.

I have lots of asparagus volunteers, presumably because one of my neighbors used to grow asparagus. Mine's not in especially well-prepared beds, it's in various garden beds that were mostly started by rototilling the lawn (well, one big one is smack dab in the middle of what passes for a foundation planting) and adding tons of compost.

It's a beautiful plant - I really love having it, and I still plan to create a special bed for it, at which time I'll move it. I've read about how that's done, and yes, I agree, fall's the best time. The ones that have popped up in places that were just too odd (like pathways) are in large clay pots, waiting for their perfect permanent place.

Since you'd have to forego mowing until fall, if you want to save the plants in your lawn, you could always try lifting it now, and moving it into a temporary location. In fall, you could lift it again, clean off the roots, and plant it in a really good asparagus bed, which, according to everything I've read, should be super-rich, very deeply dug, and ready to support the plants for a hundred years or so.