What do you mean by "(except that my Amazon Workspace is running Server 2016, and the Camera option under Privacy isn't available)"? I see "Let apps use my camera" setting in Settings - Privacy - Camera on Win2016 OS.
I don't think the setting in the Amazon Workspace (running Server 2016) under Start > Settings > Privacy > Camera matters. In my own Amazon Workspace, "Let apps use my camera" is grayed out ("Some settings are managed by your organization), but its status is "On". On a user's Workspace, this choice is NOT grayed out, but "Let apps use my camers" is "Off". In both cases, I am able to successfully use the WebCam when logging into either Workspace from a Windows 10 PC, but when I log into either Workspace from my MacIntosh, I get the "Camera not authorized" message and it doesn't work. I'm used my USB WebCam on my Windows 10 PC, and then moved the same USB WebCam to my MacIntosh. In the SystemTray on my Amazon Workspace, the FabulaTech icon in the SystemTray shows that it has detected my WebCam (both when using my Win10 PC as well as when using my Mac).
I get the "Camera not authorized" message on my Amazon Workspace when connecting to it via my MacIntosh using the Amazon Workspaces Client software. I get the message when using WebCam test web sites, such as https://www.onlinemictest.com/webcam-test/). In case it matters, the Amazon Workspaces Client software uses PCoIP, not RDP. I downloaded AMCap from their Web site, and got it to successfully display the video from my WebCam while accessing my Workspace from my Win10 PC [after following the instructions to change the default renderer (Capture -> Setup -> Preview -> Renderer) to Video Render or Enhanced Video Render]. My WebCam also works using Zoom in my Amazon Workspace from my Win10 PC. Yet on my Mac, both in AMCap and in Zoom, where a video image should appear from the WebCam, it's just a black window, yet under "Devices" in AMCap, and under "Select a Camera" in Zoom, it shows "FabulaTech WebCam" as the detected WebCam. Since everything works from Win10 and not from my Mac, it would seem that the problem is with my Mac. However, I've uninstalled both the FabulaTech software and the Amazon Workspaces Client software from my Mac and reinstalled. This is the only Mac I have, but at two of our clients (I work for a company that does IT support), two other Mac users are having the same issue.