Yes, you can do this, so long as the Bard or Fighter in question is a human or an elf, since the only condition on Half-Elven is that you chose elf or human and could have chosen the other. Any other race choice invalidates that condition—and of course, any class that doesn't have both human and elf as racial move choices also doesn't meet the requirement (and makes Half-Elven useless besides).
Half-Elven is specific to human and elf moves though, so no other mix of races are a valid choice or condition for taking the move, so no other crossbreeds work.
There's no restriction on choosing Multiclass Dabbler as your Fighter or Bard's first advanced move either, so there are no issues on that front. (Note that this is an issue if you're a class that doesn't get Multiclass Dabbler until level 6, since Half-Elven can only be taken if it's your first advanced move and level 6 is far too late for that.)
All this is subject to alteration by already-established fiction, of course. The rules give you a default and I've described that, but anywhere this description obviously contradicts already-established facts of your game, the fiction takes precedence. For example, if you have a dwarven Fighter and in the party's travels you've met half-dwarves-half-humans, then it's a fair adjustment for your GM to let Half-"Elven" be used to permit discovering your dwarf's distant human heritage. If such facts aren't already part of the game though, the move doesn't offer any help adding such facts because it's just about elves and humans by default.