Unless you are opposed for religious reasons, this is exactly the type of situation that roundup was invented for. One sprayed on dose and you can start seeding the next day. IMO, it would be more environmentally friendly than slowly solarizing a big area. You save all the benefitial insects and microbes and you can plant your wildflower crop immediately. The long term impact of a 1 time application of roundup is just about zero. All that plastic you buy to cover that much area will be with us for thousands of years. If you are absolutely opposed to roundup, you might try vinegar on a hot day. It will probably take a couple tries, but it does kill cool season grass.

I strongly agree with the recommendations to smother & WAIT. Think of this as a long-term project. Immediate results are not likely to be the best results in the long run. In my experience, weeds are one thing -- grass weeds are another beast altogether. I like the idea of using natural fiber carpet rather than plastic b/c the water can still get through. Otherwise, you can have runoff issues. If you start this in the summer, you'll have an even better chance of killing off everything. I know it's been several weeks since you started this thread, but in Dallas, we're in the middle of a run of triple digit weather, so its the perfect time for us to smother.

Most likely it is "Poa annua", annual bluegrass, This is a cold weather "weedy" grass that will die out in a few weeks. It is often used as a nurse grass in seed mixes since it does germinate much more quickly than most others. Most often, in established lawns, the seed will germinate and start to grow in late August and most all pre emergent need to be in place about 6 weeks before, so usually mid July is when to put them down.

Shot weed, Cardamine hirsuta. An annual that grows very early season and has an ability to 'shoot' its seed pods - those long skinny things at the tip of the plant - considerable distances, hence the common name :- ) And it produces seeds in copious abundance!! Ideally, you want to remove as soon as you see the foliage rosette forming in early spring - very shallow rooted and pulls up easily - but certainly once you see any flowers forming and before it goes to seed. Now all you can do is treat with any lawn broadleaved weed killer. But as it has already dispersed a huge number of seeds that will not be addressed by this, prepare to reapply several more times as they continue to germinate before the weather heats up. And it will likely reappear in fall as well.