Vertical studs only is standard practice. Any production builder would not comply with your crossbrace request because it meets code as is.
+1 BRM's advice. Add more screws, make sure they land in the studs. Noise often comes from the sheetrock vibrating against a nail (rarely a screw.)
If you put expanding foam in there it would bow out the wallboard, or pull it right off the studs.
Maybe there is a wire or pipe touching the inside of the wallboard, hitting it when it vibrates, etc. It's really not a big deal to open it up for repair and redo the wallboard. Matching the paint will be the hardest part, but because it's got edges you'd never notice any color mismatch. If the screws don't work, open up a 1 x 2 foot rectangle and jam it tight full of pink insulation top to bottom. You can watch video of how to patch drywall with backerboards on youtube, thisoldhouse, etc.
I have a noisy corner in my listening room. It is nail stiction in the house frame that squeak when house expands and contracts. It's an exterior corner that faces southwest and the afternoon sun heats it up making it expand and click every 30 seconds for a couple hours, then after sunset it cools and clicks some more. At first I blamed it on stereo components because the corner is directly behind my left speaker. After a while I figured out what it really was and now I don't even notice it anymore. Same thing with my frigerator compresor that chugs like a locomotive when it shuts off. Don't even hear it anymore, but noy was it annoying at first!!