What is the best way to add new stud framing above fireplace and tie it into existing framing?
It is up to you of course, but good solid masonry is great for fixings. I'd cut out a square of the existing drywall a bit larger than the base of the bracket and then replace it with a new piece of drywall (or tile cement backer board for even more strength) fully bedded in adhesive. I.e. instead of a small air gap behind the drywall go for full adhesive (like you would behind a ceramic wall tile for example). Fill the small seam between the new and old and allow a good amount of time to dry.  

I'd also note down where the brick joints are when the drywall has been removed (so as to avoid them when drilling your fixing points. 

The only caveat is that this plan fails if you open up the drywall and find poor rubble/ soft mortared/ old brickwork which is not very solid...

Update: Okay, if the bricks are a bit poor, you could still cut out a square of drywall and follow that with raking out the soft areas (joints mostly I'd assume) until the mortar feels solid (i.e. just scrape, don't take a hammer to it). Then mix up some render and plaster the area. Depending on the depth you'll need to do this in several layers over several days. If the mortar in the wall is lime based, I'd go with the same for the render (1 lime putty to 2.5 graded sand). Apply a couple of thin 'skimming' coats (fine sand and putty in equal measures) to the last 3-4mm 1/8th" or so to finish.