The bracket is part of the caliper. But once you've pulled it off th bike you can separate the halves. Slide the hydraulic part off the pins to split it. You could clearance th bracket at home with a grinder and a flap disc if you don't want to visit the machine shop. No one will see it down there.
Before you do, see if you can reset the caliper in a better position by pushing the piston in and sliding the calpier about on the bracket then apply the brakes. You might be able to find a sweet spot. I had the same problem when mine was new.