Plastiq won’t let you send yourself, or a business controlled by you, a check. Unfortunately. They won’t do person-to-person either.
If you can figure a way out to get the check, it otherwise sounds like a good plan. One thing to keep in mind is that 3% card has a tendency to stop giving 3%back without warning, however (e.g., GCM, GC.com)

Edit: I would look at the opportunity cost too. That's 2 new personal cards and a lot of waiting, with possibility for loss at multiple steps. It is couch MS though. It's like I did in college--buy 1 month T-bill with a CC. Too bad I had a low credit line and not super rewarding cards! Most of the gains were from interest (which was considerably higher than now.)