From a technical perspective XOGO runs great on Firestick Lite. It has been tested in the lab. However, Amazon may be blocking the "Digital Signage" category on the Lite since it relies on advertisements and Digital Signage needs to own the entire screen. We have hit-and-miss reports for Lite.
Search specifically for "xogo" and make sure spellcheck doesn't change the word. It may come up. However, we wouldn't recommend running production digital signage on the Lite since Amazon can be inconsistent and reserve the right to block any app for any reason at any time.
We love the Fire products for the most part. XOGO was tuned to run well on them. That said, it's up for debate if Fire should be used for serious digital signage since it's default mode upon a power failure or reset is always a streaming media player compared to the XOGO Mini which is single purpose digital signage appliance.