LLMs have some amount of reasoning capability. If you tell them, they will get rate limited after 10 searches. Hopefully they will put that reasoning to work to calculate out the 9 searches that will get them to where they need to go.

If you hide this from them then they will likely tell their user that there was an error searching your website and move on to another one.

I suggest trying to be transparent with your rate limiting, perhaps not the punishment durations, but at least how many searches or function calls they have left.