I think you have to let the music/song traditions develop on their own.
If you have a creative marketing team, or, as the White Sox have, a pretty savvy ballpark organist (Nancy Faust), they try things out that catch on and become your ballpark traditions, and then are co-opted by others (e.g. "Sweet Caroline," which started with the Red Sox, I believe).
Nancy Faust started playing the chorus from an obscure early 70's hit when opposing pitchers were being changed. This chorus (Na Na, Hey Hey, Goodbye) has been co-opted across the country and in just about every sport. Its a big part of White Sox games today, and we still use the ballpark organ version, rather than a recording of the pop hit itself.
Another favorite for us back in the 80s was after doing "Take Me Out to the Ballgame" for the 7th inning stretch, Nancy would start her own version of "Run Around Sue" with the fans singing along to the "Woe Woe Woe" part. It was great fun. Nancy is in semi-retirement now (she only performs for day games and weekend games), so we get more "canned" music at the games. Some of this works, and some it doesn't.
For example, back in the 90s, the White Sox would always play "The Boys are Back in Town" when the team took the field after for the first game after a road trip. Cute, I suppose, but it got old.
I agree that winning really helps cement these kinds of routines. After the turn of the century, the White Sox started a tradition of playing a montage video of highlights from White Sox history on the big screen in center field at the start of every game as a prelude to introducing the starting lineup. It uses movie soundtrackmusic (part of it from Pirates of the Caribbean, I believe). I still get chills and a lump in my throat every time I see it, as the video clips represent nearly a century of White Sox baseball, and almost every clip was something I saw or experienced. Here's a link to a recording of it from the 2006 season:
Here's a "bootleg" version of the same thing from 2007, to give you the full affect. After the history montage, they switch to heavy metal "Thunderstruck" with lightening bolts and stuff, and introduce the starting lineup. These home movies don't do it justice -- it is a raucuous good time. Quality not good but you get the idea:
We've done it this way since 04.
The "Thunderstruck" song has also been sort of the official opening song for the White Sox since the mid 90s. I prefer Nancy Faust's organ music, but it has become a White Sox tradition.
Another nice tradition is when we win the game, the fans exit to the music of the Blues Brothers "Sweet Home Chicago"
You'll develop you own traditions in due time. Hopefully your marketing guys will be creative, and find a few that really work and fit in uniquely with the Nats.