Definitely just directly into the composted manure. Our very first asparagus bed planted 4 years ago, they were 1st year crowns were just planted directly into the bed completely only of composted horse manure.
We have 2 beds that were started out as seeds and they too are doing brilliantly as they were also planted directly into composted horse manure.
Think you need to plant them out sooooooon, as they might not have time to establish before winter comes.
As for your current beds, think they will do brilliantly once you add some composted manure. We had asparagus growing out of our ears this year! ha!
Remember…. no dig! 🙂