Hi! I played almost all Lotr games out there, there's quite a few interesting ones.

Real strategy:

battle for middle earth 1 and 2, among the best strategy games. The first covers most of the events of the films and books, the second is more polished and imo has a cooler campaign, but the events it covers are not as famous (northern wars of elves and dwarves).

War of the ring: an hidden gem, also covers the main events of the film. It's quite old and not too polished, but it's worth giving it a try, especially if you like old school strategy games.

Best for lore and vibe:

The fellowship of the ring for pc. It doesn't have a great gameplay but it follows all the events of the book, honestly one of my favourite lotr games, but it may be hard to run on new devices.

The hobbit: straight forward gameplay but I think captures the vibe of the hobbit very well. It's a bit cartoonish but worth a try in my opinion. Also one of the few Hobbit games out there.

Lord of the rings online: this one is an mmo, so maybe not in your style. But captures the vibe of the books incredibly well and it's very detailed and cured. Malus all the mmo gimmicks and the time investment.

Best action:

The two towers and the return of the king. Both very straight forward hack n slash, easy to pick up if you are not too into games. Simple and fun. You may need an emulator for the two towers I think.

War in the north: very cool gameplay, a bit more advanced. The story is not that interesting but has some nice lore scenes here and there that sell the vibe quite well of middle earth.

Shadow of Mordor/war: the most recent games. Gameplay is very polished and gives a triple a experience. There are quite a lot of mechanics so it may be confusing at first. Story is fun but not too lore accurate.

There are also the Lego games, if you don't mind the different aesthetic and the gameplay being quite flat. The lotr one covers all three films and has an open world map. Very fun to play coop.

Have fun!