I went through all the entries for hotkeys in the registry. These are the ones I found, setting the values to "None" disables the hotkey. Copying the following into notepad and saving it as a .reg file allows for easily disabling of them all.
Windows Registry Editor Version 5.00
[HKEY_CURRENT_USER\SOFTWARE\AMD\DVR]
"MicPushToTalkHotkey"="None"
"SaveInGameReplayHotkey"="None"
"SaveInstantGifHotkey"="None"
"SaveInstantReplayHotkey"="None"
"TakeScreenshotHotkey"="None"
"ToggleCameraHotkey"="None"
"ToggleDvrRecordingHotkey"="None"
"ToggleMicrophoneHotkey"="None"
"ToggleRegionRecordingHotkey"="None"
"ToggleRsPerfRecordingHotkey"="None"
"ToggleStreamingHotkey"="None"
[HKEY_LOCAL_MACHINE\SOFTWARE\AMD\DVR]
"MicPushToTalkHotkey"="None"
"SaveInstantReplayHotkey"="None"
"SaveInstantGifHotkey"="None"
"SaveInGameReplayHotkey"="None"
"TakeScreenshotHotkey"="None"
"ToggleCameraHotkey"="None"
"ToggleDvrRecordingHotkey"="None"
"ToggleDvrToolbarHotkey"="None"
"ToggleMicrophoneHotkey"="None"
"ToggleRsHotkey"="None"
"ToggleRsPerfRecordingHotkey"="None"
"ToggleRsPerfUiHotkey"="None"
"ToggleStreamingHotkey"="None"
Some stubborn ones still remain though, such as toggling the FPS counter, so I had to roll back to a previous version to get rid of all of them.
Thanks for ruining my day, AMD.