Meloxicam is an anti inflammatory and pain reliever. From what I know about it, you have to administer quite a lot of it so maybe the food route is better. And it’s good tasting.
Make sure you give that in the morning, not at night so she can be comfortable during the day.
She may have eaten fast and got some egg stuck in her throat. Or the meloxicam is sticky so she may be feeling that in her throat.
Good news, Meloxicam has no egg withdrawal time. You can eat her eggs. I’m not sure why the doctor would have told you you couldn’t. It’s chicken ibuprofen.