Neither one is particularly well located. I believe that Viking docks on the Pest side, I know others whe were taking Viking had made that statement. Even if Viking did dock on Buda side, Buda is a very, very, very small part of what there is to see and do in Budapest. Sure, if you have a week and don't mind making the commute from Buda it might be okay but I don't think it is particularly convenient if you want to maximize your time in Budapest. A good Budapest visit is a full three days. That means three days other than arrival day and departure day. If for some reason you must pick one of those two I would choose the Bellevue hands down based on location but I haven't stayed in either so I can't give you any better advice. The only hotel i have stayed in in Buda was the Art o'tel and i wouldn't recommend it. If i were to stay in Buda it would be at the Gellert; but that just for the history and the bath house. I understand its not a "great" hotel.
IF, however you have more latitude then I suggest you look on the Pest side. I think the most advantageous area to stay in is along the M1 line. Go to Google Maps and enter "Oktogon Budapest, Hungary". This will take you to a major intersection on Andrassy ut (Avenue). Now draw a mental line from the Oktogon down the center of Andrassy ut and beyond until you reach the Danube river. You want to find a place that is maybe 2 blocks above that line or three blocks below that line. If you now type "Hotel" in the search window you will get markers for hotels in the general area. This is how i do it every time i go someplace new. I think the best all around hotel in the zone is the K&K Opera. If that is too expensive for you look at Opera Garden Hotel which gets good reviews. I've only stayed in the K&K and across Andrassy ut at the Casati; both were very nice and both ideally located. You also might think about renting an apartment in the neighborhood.
If you come up with some other choices let me know and I will tell you what I know about the neighborhood and ease to reach the sights.
Hope this helps a little.
Here is a sort of top 11 sites of Budapest. Test your accommodations location with Google Maps. Put in your accommodation address in from and the BOLD text below in to and then check the walk and metro times and distances. Also look for accommodations that get you where you want to go fairly quickly on foot or tram vs. the underground. The one exception I would make to that is the M1 which is in and of its self an attraction.
1. Castle Hill (address for the lower station of the funicular) Budapest, Clark Ádám tér
2. Danube Cruise Budapest, Vigadó tér
3. Hungarian State Opera Budapest, Hungarian State Opera
4. Central Market Hall Budapest, Central Market Hall
5. Sechenyi Bath House Budapest, Széchenyi bath house
6. Chain Bridge Budapest, Széchenyi István tér
7. Parliament Building Budapest Parliament Building
8. Andrassy Avenue, Budapest, Andrassy ut 30
9. Dobb Street Synagogue, Budapest, Dohány Street Synagogue
10. Vaci utca (sort of a mid-point of the best of the street), Budapest, Piarista utca 6
11. St. Stephen's Basilica, Budapest St. Stephen’s Basilica
For evening dining and strolling the best options are lower Andrassy ut (use the Opera House as a point for distance), Vörösmarty tér and the nearby Danube embankment, and courtyard in front of St. Stephen’s Basilica. There are others like Kiraly utca, but these are my favorites.