Hotel des Balances or Hotel Schweizerhof Lucerne

I will comment on location, not amenities - I haven't stayed in any of the three.
I find that des Balances and des Alpes have the best locations, overlooking the river - whereas the road in front of Schweizerhof is a busy one; I found the location less charming in my July visit.
And between the two, I would pick the one where you can afford a room with a view.

Alison you can also look into staying in a town close to Lucerne at Hotel vitznauerhof which is a quick 10 minutes ferry ride.

https://vitznauerhof.ch/en/

My second choice is Hotel des Alpes. It is convenient to public transport.

We stayed at Hotel des Alpes in May. We were able to walk very easily to and from the train station. We splurged on a room with a view.

I vote for Hotel Des Alpes- we loved it there! In addition to a room with a view of the lake, see if you can get a room with a balcony AND a view of the lake.
Safe Travels!