I've never seen a quick-connect valve with a mounting bracket; they all have a female pipe thread inlet and are supported by the pipe. As Dale showed, Escape appears to be using a pipe thread to flare adapter
to connect it to the copper tubing, and strapping across the flare nut; they don't use a bracket at all.
I would use at least a short piece of pipe (a nipple) or a through-bulkhead connector between the valve and the flare adapter, and use a bracket holding the pipe or bulkhead connector. It would require one or two more pipe thread connections, but I think it would provide a much more solid way to hold the hardware.