Do you think it would be a good idea to give her a bath to try and eliminate some of the insects?
I don't honestly know. My gut is to say it would be better to pick them off rather than actually giving the bird a bath. If you're extremely careful about water temperature and making sure she's in a warm, dry, still place so that she doesn't get chilled, it might be okay to do.
Like i said, there was one but it had to shut down a few years ago due to a lack of funds. There is still some evidence of it online but it is no longer open
I'm assuming this is the one you're referring to, but they have posts from about two weeks ago about a bird they're rehabbing?
https://www.facebook.com/Cayman-Wildlife-Rescue-255934667709/
Some of these groups might help or at least point you in the right direction:
https://caymanresident.com/live/pets/lost-and-injured/
EDIT: From the above link:
If you find an injured wild animal you can take it to Island Veterinary Services who will nurse it back to health. They appreciate donations for this service (which they offer for free). They can be contacted on (345) 949 0787. They don’t however, have enough staff for pick-ups.