I would honestly just make two expressions, start bracket and end bracket, and then assign them to hot keys (like 9 and 0) for easy input. That way you can adjust them easily around any object. 

Shape designer would allow you to drag them, but once you drag it, it "overrides" the original with your modification so that if you have the shape again elsewhere in the score and drag it differently, it will change it everywhere in the score.

You could also try making a custom Smart Shape, consisting of a line of zero width with the brackets as text at the left and right ends of the "line". You'd enter it just like the hairpin and (possibly) set the anchor points at the same locations as those of the hairpin.

I just had that thought! Thank you, both.