I think you would be able to. I’ve used TinyCam on one of my Fire TVs before, but didn’t try it for serving up the streams to my dashboard.
That being said, the TinyCam developer specifically recommends the NVIDIA SHIELD TV as a ‘server’ for TinyCam. I have a few of them and they are amazing devices, so it doesn’t surprise me.
Edit: @Jeff_Fourman switched over to an NVIDIA SHIELD for streaming his cameras and said it works really well for him too.