I would say it's
Alkmaar, Netherlands
I'm a cheesy city
One of the greatest cheese markets in the world is in Alkmaar.
Containing auk but in the main language of the country I'm in.
Thanks to @LordofDark and @MichaSprengers: "auk but" in Dutch is "Alk Maar"
I have a sports team that got crowned twice.
The local football team AZ Alkmaar won the national championship twice, in 1981 and 2009
Also placed within me is a museum dedicated to a band that once tried to remake The Lord Of The Rings.
There is a museum dedicated to the Beatles, who unsucessfully tried to adapt The Lord Of The Rings.