You can use keywords to select search engines in both Chrome and Firefox. If you configure "custom1" as a keyword, then just type custom1 followed by a space into the address bar it will activate that search engine and send the rest of your query to that URL.
This wouldn't let you use multiple "flavors" of Brave though unless they expose different URLs for 'Goggles'; sounds like you just are allowed to enable one as a sitewide setting.
They do have different URLs for different goggles. Do a search for the word "test" with a Goggle on, then grab the URL from the address bar and paste that into your new bookmark. Replace "test" with "%s", add a keyword, and you're good to go. Repeat with other goggles.
