Vlad I don't have the rate limit screen right now, but I think the dog's talk balloon is a very good place to work with. You can put some additional explanations and add the search button at the end.
About the button itself, I have two ideas:
A single button with "search with " and an input box where they can choose an alternative search engine. Take the "Region" button as model, there is a default region and you can change to any other region you want and save it. I think this is the easiest and best solution as it will be just another user preference like "Region" or "Safe search".
Adding multiple buttons side by side (maybe with a horizontal scroll) from multiple selected search engines. Sorting the buttons by the last click would be useful to avoid complaints about how their favorite search engine is far to click. Maybe leave an option for the user to add an extra button with a new search engine manually or select from the bang list too?