Just about every mass-produced ice cream contains carageenan, which would not be permitted.
Perhaps try making your own? This way you know what goes in to it http://www.lemontreedwelling.com/2016/07/no-churn-cherry-vanilla-ice-cream.html
Ducktrap River in Maine has smoked fish without sugar -- but which ones escape me at the moment. Their Kendall Brook Salmon comes to mind but I don't have any around at the moment. Will try to snap a photo when I'm at the store next.