I think that it would probably cause a bit of concern for your readers; I think us sci fi and fantasy nuts are sticklers for fact or authenticity and if your narrative does not explain the origin or appropriateness of these two names, then it may bring the reader out of the zone.
I'd be relatively forgiving of La Hainne as a name, regardless of the french not figuring, although I may be tempted to make it one word or change La to Luh or Lah.
Having said that, I am the person in the cinema, or book reader, who never ever
sees something coming as I am so engrossed, no matter how strong the foreshadowing is. You'd have to shoot me with Chekhov's gun before I event went 'ahhh!'
so the likelihood of La Hainne taking me out of the playworld would be miniscule. Tolkien on the other hand.....
Why are these names- esp Tolkien - so important to you? Would it cause you grief to rethink them?