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.