Regarding hosting the game client on AWS. Wouldn't a virtual machine with virtualized graphics card driver work? Windows instances are quite pricey, I suggest checking out if the client would work through Wine (or Proton). As far as I remember, AWS also has GPU instances, but again, they are not so cheap. If you really want to run it on the cloud while failing to use an emulated GPU, but otherwise you get it to run on Linux with Wine or derivatives, there are open-source projects which would allow you to mock the graphics library the game is using with simpler functions (say not using GPU drivers at all) and this way you could do everything without a GPU, even in a docker container.
You might also want to read about virtualizing the GPU in qemu or virtualbox.