Being a language guy, I'm wondering if you/we can come up with a good term that expresses "to search the Net using the Kagi search engine". To me, not using Google anymore involves pride and I'd be happy to show that pride (even at the risk of coming across as a snob). Saying "I will ***** it" instead of "I will google it" will no doubt also raise questions by others who are not aware of Kagi, and therefore also provide a good opportunity to promote it.
I really like the name "Kagi", but as a verb, it seems limited. "To Kagi" sounds fine, but once you start telling someone you will "Kagi it", it already turns into a tongue-twister. Not sure about other languages, but I do know that, in German, "ich muss dass kagi-en" really doesn't work at all.
I also read that the Kagi mascot is called Doggo FOR NOW. I understand it has probably become used to that name at this point and responds to it, but knowing dogs, I'm sure if it is rewarded with enough bones and other treats, it can learn a new name in no time. So maybe it is time to let Doggo mature into a character with a more unique name, and then something from that namespace could be used as a verb.
An alternative approach could also acknowledge the fact that "Kagi" means "key" and go from there.
Just throwing out some thoughts, I understand that this idea oversteps the boundaries of function and takes a step into the forbidden garden of concept, but maybe it resonates with something that was in the books already.