Grog_280
Though I don't know if it will make a difference, something you can try on your Fire TV is to go into Settings > Preferences > Data Usage Monitoring. Turn it on, then go to Set Video Quality and change it to something lower. By default it's on Best. If you change it to Good, it may help to reduce the resolution at the source by only allowing the TV to use an amount of bandwidth that will trigger that to happen.
With this said, if the problem is due to system congestion, or a combination of congestion and the high latency inherent to this type of satellite internet, it likely won't make any difference. Congestion tends to be most prominent in the evenings, when the highest number of people are online and many are streaming. As for the latency, unfortunately, the high latency can sometimes cause issues with streaming with some services, even when the overall speed is good enough.
I'd also check to make sure that nothing else is using the service at the time you're trying to stream, sapping available bandwidth.