I’ve been taking Meloxicam for many years. It’s never put weight on me. I don’t believe that is a side effect either. It’s the menopause not the meloxicam.
May be wise to have your labs checked Creatinine and BUN and GFR for kidney function. This medication can harm kidneys and that could be a sign if fluid retention weight gain
I also have high blood pressure and take Ibuprofen (which is like meloxicam as its an NSAID), I was told to drink MORE water so that the Ibuprofen doesn't make the blood pressure worse. Its some sort of chemical thing, a pharmacist told me once. Ever since then, my blood pressure has been so much better. Good luck to you.
Hopefully it's mostly "water weight" and disappears quickly. I'm really not sure though.