That's the trademark side of the case, and yes, In N Out could probably successfully argue that usage of their logo implies endorsement and/or is likely to create consumer confusion and that it's therefore a trademark violation. That's pretty easy for DoorDash to put to rest; just stop putting In N Out's logo on the site. A few good, highly-visible disclaimers would probably put any remainder of a trademark claim to rest.

A modified menu would be tougher. You can argue that there may be copyright infringement, and that may be valid to a limited extent, but DoorDash could put up a version of the menu that wasn't copyrighted, as its heart, menus are factual arrangements, and facts are not original creative works subject to copyright (though a specific arrangement of those facts may be). In N Out could also claim that there's trademark infringement if DoorDash is using trademarked food names, but I think it'd be a harder argument to win, since DoorDash can argue it's a nominative use.

IANAL.