I too want a shortcut to easily conduct my current search with another engine.
But I note it is pretty easy with no changes. Just type "/" and you are refocused to the search where you can now just add "<space>!g<enter>" (or whatever bang you like)
Still....
How about you provide "!" as a keyboard shortcut?
I see 3 flows/UXs:
a) a new kagi.com search tab is opened, prefilled with current search terms followed by a "!" and cursor positioned there
b) user is prompted (modally) for a bang which on <enter> conducts the provided bang search optionally in a new tab (if shift key held down during enter)
c) focus is switched to search bar, with a "<space>!" gratuitously added.
PROS/CONS
a) would be somewhat jarring, but works with assumption that user will want to return to original kagi unbanged results.
b) introduces a new modal dialog which asks for support issues. Also, I know you wont like the default (unshifted) behavior is to leave kagi.com, but it would be consistent with just adding a bang to the search box.
c) is quite natural. Combined this with <shift><enter> in the search bar opening a new tab for the requested search (whether banged or not), and I think you cover all needs. Perhaps only lacking would be <esc> in the search bar to delete any trailing "<space>!\w\s$" already entered and return focus to previous element in focus, allowing user who changes their mind (or who inadvertently hit "!") to return to where they were.
Nice UI problem! I hope this helps!