Assuming the hole is only about 1/4”, remove the shingles, install a 2x4 brace between rafters beneath the hole, put a 6”x6” square of Peel ‘n’ Stick on top of the underlayment or double up with another underneath if you can access it, install new shingles, use a stainless steel lag screw, drill a pilot hole, slap on some roofing caulk, mount the bracket with the lag screw, and then apply a little more caulk on top. Done!
If the hole is any larger than a 1/4” or it’s soft around the surrounding decking, then I’d probably cut out and replace that section of decking. You can pick up a 12” x 12” deck patch from a roofing supply or you could install a couple of additional braces between rafters and do it that way.

The brace between rafters will provide a stable base for mounting the chimney support leg and needs to be installed if you are going to use the same location for mounting the bracket. While you are up there, remove the other drywall screw on the second chimney brace and put a stainless steel lag screw there as well and make sure it either goes into a rafters or you install another brace between rafters.

Thanks Tony. I forgot to mention that it can't be accessed from underneath (at least, not without making a giant mess). Reason being it's a sloped ceiling inside so there is no crawl space. How big a deal is it to remove and replace the shingles? I've never done roofing work before and I'm not sure what brand/model the shingles are so I don't know if I could get ones that match perfectly. Or are you referring to replacing ALL the shingles? (Thus re-roofing the entire great room)?

That complicates matters a little then. If you don’t have any roofing experience, then I’d recommend you hire a roofer or a competent handyman to patch and relocate the chimney bracket. It might be a simple repair, but if your roof is leaking in that area, chances are it’s a little more complicated. You don’t want to have the shingles off and holes cut in your roof wondering how to proceed with a couple of days of rain breathing down your neck.
If you do want to try it yourself and it turns out to be an easy repair, I was thinking you likely need to remove just two shingles. They’re easy enough to remove with a prybar. Just check out a few YouTube videos. Depending upon where the hole lies, you may have to remove three or four. If you don’t damage them too much, you might be able to reuse them, otherwise Lowe’s and HD carry shingles that are a close enough match to most other common types and brands. Take an old shingle with you and someone will help you match it. Before you patch the hole, bend a bit of coat hanger wire in a 180 and push it into the hole. By rotating it, you can find the exact location of the rafter and then mark it on the roof. Patch the hole, replace the shingles, relocate the bracket to a rafter with the proper fastener. Done!
A word of caution though. What seems like a simple job from the top of the roof can turn into a complicated process as soon as you start to see what’s under the shingles. If you can poke around the location before you remove any shingles you may be able to glean a little more knowledge of what to expect. If the roof is “soft” and spongy around the bracket mounting point, then it’s likely you’ll need to replace decking. That’ll involve circular saws, decking, layout tools, nails, integrating underlayment and likely a bundle or more of shingles. Definitely let the pros do that for you.
If the roof is firm, then it’s likely that the job will be simple and straightforward, but don’t bank on it.

OK that sounds like good advice, thanks Tony. I'll get up there and poke around at it a bit. If the decking is soft or spongy as you say, I'll probably end up calling a local roofer. The last thing I'd want to do would be to make it worse, and unfortunately I have a history of doing that. Ask me about my leaky water softener, LOL.