I think I found a solution, and it's thanks to you guys.
I'm going to use my MLD Ultralight Hammock Tarp for this, using the bike as a "pole" on one end and then guying out the other 3 corners. Since I have Ortliebs, I only need room for me.
Now, I think this tarp is a little small for a serious storm (I had concerns about that when using it as a hammock tarp, too, but I ignored them). If I set up like this I will likely get some wind spray.
To supplement it, I'm going to use a water resistant bivy bag. Since it's cool at night in the desert, condensation shouldn't be a problem.
Ideally I'd go with MLD's Ultralight Bivy, but ron tells me there's a (perfectly understandable) 6 week wait time on one. Instead, I found the North Face Simple Bivy.
I don't particularly like North Face anymore, but this bivy looks fine. It weighs 16oz, has a bug net over my face, is highly breathable and water resistant (not waterproof) and should serve my purpose just fine. With some haggling, I got it for $99.95 with free shipping and free returns.
If it doesn't look good in my backyard, I'll probably grab something like the MLD Patrol Shelter or Trailstar since they ship the next day.
Thanks for the help! I'm a tarp camper now!