Torvehallern Food Hall, Frederiksborggade 21, has wonderful gf options. There is even a paleo restaurant there, which always means gf. There is a lot of healthy food available- fresh salads, etc. Fun place, as well.
We ate at an Asian restaurant in Tivoli, which had gf options. It was pretty good, much better than the guide book led me to believe. Definitely a cut (several cuts?) above “amusement park food” and the price, around $50 U.S for two in July 2023, seemed reasonable.