I have this idea for a variant on Eldritch Knight that might just turn into something else entirely, but for now that's where it's at. One of the things I'm changing is instead of cantrips giving the class to a smite. Force instead of radiant. Is there a way to remove cantrips as options from the spell list? Preferably without adding every single spell by hand.

In DDB, you'd have to create a new Fighter subclass based on Eldritch Knight and modify its Spellcasting feature to remove its cantrips/set its Cantrips Known to 0.  I don't know how to do that offhand as I've never monkeyed with that aspect of the magnificently clunky and unintuitive homebrew editor, but I'm reasonably certain it could be done.

In a pen-and-paper tabletop game, you'd just tell the guy not to record any cantrips.  I would be careful about replacing cantrips with Smite, however; EKs get very few spell slots to Smite with, and half the power of Smite is upcasting it to gain more damage, which Eldritch Knights also can't do very well.