If you care to design your own style sheet, then you can modify a lot of things like this by digging down into the Core.nb style definitions. The Suggestions Bar definitions are under: Styles for Mathematica System-specific Elements, Predictive Interface Styles. You could copy the appropriate definition into your style sheet, change the FontSize and probably delete the other entries, which would be inherited.

On the Mathematica page at my web site I have an extended essay that also discusses style sheets.