I use a small Mackie VLZ402 mixer for that purpose, however, it only has bass and treble controls. Not exactly cheap either, and you'll need a mixer which accepts DI. I think you'll find it difficult to find something with bass/middle/treble, headphone output and a 'clean' sound for low cost. This takes you into the bass headphone amp territory, none of which are cheap and some which add their own character, which you don't want. I also use a Phil Jones Bass Buddy, which again is clean sounding and has a 5 band graphic. These are unfortunately expensive nowadays.
Hope someone can throw in some alternatives to consider.