Good question. I honestly have no expertise here, but I have a tiny bit of experience. Soil is generally considered a living thing but most bagged compost has holes in it, so I think it would be fine. I bag my own at a local gardening store and we use these reusable transparent bags. They have no holes in them. I had mine bagged up and behind my house in the 90+ºF sun for a couple of months before I used them and had no trouble other than having one of them stink a bit. Still loose, still with nutrients. I wouldn't worry much about it, especially if the stuff is breathing.

A second alternative would be to create the beds this fall, and let them "mellow" over the winter and be ready to go in the spring. Many lasagna gardeners built their "new" beds in the fall for use the following winter. You could put down cardboard or a thick layer of newspaper on the spot you want to build the bed, then just throw the Mel's mix on top. By the time you were ready to plant the following spring, the newspaper and grass under it should be fairly broken down. (Actually, I would suggest adding a layer of shredded leaves into the mix as well. The finer the shred, the better.)