DISCLAIMER: Check your local laws before installing this modification. I am NOT responsible if you get a ticket/summons or cause damage to your license plate/car for following the instructions on this page. Try this modification at your own risk.
I purchased a Team-Miata Swinging license plate frame awhile back & it bent & became useless a few months after purchase. Basically, it's just a flimsy door hinge with a piece welded on which fits the license plate. You can do much better by going to Home Depot or Lowes & purchasing a stockier 4" door hinge yourself. You'll have to drill holes in your plate to mount it to the hinge. Also, you don't mount the plate to the hinge at the top or bottom like it's a door. You have to mount it in the middle someplace. Just fiddle with it by attaching the plate to the hinge with a piece of tape until you get the balance just right & then drill the holes appropriately. You'll have to take into account the weight of the screws your using as well for the balance line.
I have it mounted at the top of the "mouth" b/c I've had to remove all of the support structures from the bottom to accommodate the turbo intercooler. You can mount it on the bottom of the "mouth" if you still have all of the support structures.
This whole project took me little more than 45 minutes even with all the trial and error. It should take you ways less time as I had to figure it out & experiment.
Crude drawing of the Team Miata hinge. It isn't to scale but you get the idea. What makes the Team-Miata hinge so expensive is the fact they weld a piece onto the hinge which fits your license plate exactly.
Side view of the US$1.35 4" hinge I got from Home Depot. It is MUCH stouter than the one from Team Miata.
View with the plate flipped down as if at speed. "At speed" is ~15mph. I've seen it for myself on the sidelines from auto-x when a friend drove my car. You can also see the places where I drilled mistakes. I kept having to go more towards the middle of the plate to get the balance right. But, even then, to make the plate go down even easier, I used a few stick on weights they use to balance wheels with. Also, if you'll notice, there is the remnants of the old license plate bracket on top. The weakness of the Team Miata/hinge is it only has one mounting point. Over time, this one mounting point gets weak & bends. I took the stock mounting bracket & cut the front ends off. I mounted the hinge & then mounted what was left of the original bracket over it to help keep the hinge from being forced downwards from the force of the wind at speed.
The view from the front of the plate. Even with those extra holes & the bolts in the middle of the plate, I've never been stopped by a cop for it & you can still see everything on the plate just fine.