A typical principle for font pairing is contrast. If there's a sans-serif in the header, using a serif for the body copy is a great way to provide contrast.
However, I don't think that principle applies here. The typeface in question is so funky, so quirky, and has so much style that it would probably benefit from a cousing that's a lot more simply drawn.
I did a little bit of poking into Bauhaus the typeface, and I came across this very interesting article. "Bauhaus"-like typefaces are based off a drawing by Herbert Bayer, a member of the Bauhaus:
My initial thought was to look at some other Bauhaus typography to complement the Bauhaus font, but as the article points out, this wasn't typical of Bauhaus typography. But looking at the drawing strips away a lot of the funkiness and makes it clear that it is definitely a geometric typeface, and it fits in with the constructivist style as well.
So, if you're looking to keep it "in the family" but provide some differentiation, I'd recommend you try Futura or Avenir. Twentieth Century MT and Century Gothic are lesser alternatives, but you might have those available on your computer already. The Fonts In Use article has a nice list of "Geometric, contructivist typefaces based on the design ideas of the Bauhaus" at the bottom if you want more ideas. Some of those would be really interesting choices for secondary headers.
Here's Bauhaus 93 paired with Futura: