I would use 308/307 over 301/302. Semantically, using a 303 redirect might be the most appropriate signal for what they are doing. They could redirect to a sp

I would use 308/307 over 301/302. Semantically, using a 303 redirect might be the most appropriate signal for what they are doing. They could redirect to a specific result page if it's a clear, unambiguous match and they could redirect to a search results page if there are other possible matches.