I'm not sure how it works for kagi, but for my searxng instance I use: '(.*\.)?instagram\.com$': 'flufi.me/profile'
It might not work for all Instagram url's, but so far it's been ok for profiles I've looked at and no "sign in to see more" like you get from Instagram.
Then I've also got some domains I remove from results and some that I give a higher or lower priority (in my case, for example, I try to give zeldadungeon.net a higher priority when I'm searching for Zelda stuff. Nothing against the other sites, I just prefer this one)
(and Kagi, I'm not trying to refer your users to searxng and actually saw that there's issues/pull requests for kagi support in searxng where users would use an API key for paid account)