Thanks Dan, that's excellent guidance.
I ended up pulling out the entire vent and just fixing it on the bench. It was a little finicky, but anyone with a bit of patience can do it.

It's basically a matter of popping off the trim piece under the vent (flat trim tool or a butter knife works well) and removing the one screw that holds the vent in place. Then use a pair of small hooks to engage two holes in the bottom of the vent (just below the lowest vent slat) and slide the unit up and out.

Once out, you can either try to work on it like that, or take it apart further. There's a black plastic ring around the perimeter of the vent that you can pop off by working your way around the vent releasing the locking tabs. Then the vent slats easily pop out and you can quickly reattach the vent tab and reassemble the whole thing. Slide the unit back in, and replace the screw and trim piece.
Although expensive to just fix a tab, you could purchase a new vent assembly for about $35 and replace as described above in about 5 minutes.

Finally, that little metal piece in my photo above actually drops down into the tab so the little metal hump faces out. The hump acts as a "bearing" of sorts to ride against the front edge of the vent slat once the tab is reassembled back onto the slat.