... cut the angle corner tape joint and the ceiling to wall tape joint ...
Sorry Rob but this I would avoid like the plague. The adjoining walls and ceiling have textures of their own that you don't want the patch to affect. Doing as you suggest would require new tape and mud on those planes. Getting the new tape and mud to match will be impossible. A normal person might be able to stop looking at the mismatch but MDS might be forced to tear his whole house down. (sorry, couldn't resist)
My suggestion is to add blocking/backing (as you mentioned) across the top in between the studs and on the sides. The blocking can be 2x4 or plywood strips or metal and only needs to be attached to the existing drywall. It doesn't have to be attached to the studs to have any structural capabilities. Keep the extent of the patch/mud work confined to that wall. Add whatever insulation before the patch. Have you're drywall finisher use E-Z Sand 20 or 45 for the tape and first coat (which happen at the same time). He can go out for a few beers and come back for a 2nd coat that day. If he doesn't drink, don't trust him. He could snap at any minute. It should take another coat or maybe two coats to get it ready for tile.
If the wall has a heavy texture on it he can knock down the high spots with a quick sanding and skim the whole thing out with regular mud in like 20 or 30 minutes which will make it easier for the tile guy. Chances are pretty good that your tile guy can handle this or will know somebody who can. I'm thinking $400-$450 will get that wall ready for tile including material. It being Texas means you can probably get it for half that and this economy probably means you can get it for $175 but as TLS says, your most expensive purchases are the cheap ones.
If it was me I would yank out the fire place. Texas, right?
... and I wouldn't let Jamie near any power tools ... ever.