Actually the Holodecks allow for crossover as an integral part of the setting, that's what they're there for! So sure, there could be a "Lord of the Rings" Holodeck program,
There could be many things, but not Lord of the Rings
. Copyright law and all that, and the current license holders of LOTR film material are fairly litigious (they wanted to shut down a pub because it was called the Hobbit).
I could see a Star Trek series doing a holodeck pastiche of Lord of the Rings, in the same way that Dixon Hill takes his cues from Philip Marlowe and Captain Proton is basically Flash Gordon, but not actually use Tolkein's legendarium.
You could called it Generic Standard Fantasy otherwise known as "Dungeons & Dragons". the typical Generic Standard Fantasy involves an Earthlike planet with elves, dwarves, gnomes, halflings, half-elves, and humans, typical evil races are orcs, goblins, hobgoblins, gnolls, lizardmen, trolls, giants, dragons, and the undead are skeletons, zombies, wraiths, ghosts, vampires, liches, ghasts, ghouls. Typical standard characters are fighters, rangers, wizards, clerics, rogues, paladins, bards, druids, sorcerers. Those are your basic cast of characters that would appear in a Generic Standard Fantasy Holodeck program.