Could you please specify which tracks from The Lord of the Rings you like the most? [B] There are quite a few of them in total.

In my opinion, the main thing here is not to use the main musical themes with motifs that are a hallmark of the work. In The Lord of the Rings, these hallmarks are the following tracks for me:
Howard Shore - The Breaking Of The Fellowship
Howard Shore - The Bridge Of Khazad Dum
Howard Shore - Minas Morgul
Howard Shore - A Storm Is Coming (if you listen from the very beginning, you might not recognize it, but at 00:32 everyone will understand what it is and where it's from)
Howard Shore - Into the West

And these ones are quite suitable:
Elven city: Howard Shore - Lothlorien and/or Howard Shore - Evenstar (I removed the "shamanic dance" sound effect in the elven city and replaced it with the in-game sound of waves for Evenstar)
main menu: Enya - May It Be
for battles (you choose for yourselves): Howard Shore - A Knife In The Dark and/or Howard Shore - Amon Hen

and there are many other tracks.

The same applies to the Game of Thrones mentioned above - using the main themes is not a good idea because it's like advertising the series - we shouldn't be thinking about another project - it pulls us out of the atmosphere of the game. There are many interesting things in Game of Thrones, but the main theme should definitely not be used (to be honest, I once put it in the dwarf city, but that was in the past :D)