It is possible with a settings change! Just run
gsettings set com.solus-project.brisk-menu label-visible false
Edit: I’m using Brisk Menu 0.4.5 from the PPA in 17.04, which usually has .3.5, and I’m not sure when this feature was added. Artful/17.10 also has .4.5, so it will definitely work with 17.10.
You also change the label by editing label-text, so for example to make your menu say “OMG! MATE” run
gsettings set com.solus-project.brisk-menu label-text 'OMG! MATE'