Using the --installation-id flag, I was able to push my own custom app to my Tidbyt without having another app in rotation. Are you unable to remove the unwanted app once yours is pushed? (Note this flag is documented here: Tidbyt | Dev)
In terms of push rate, I will say that I was getting issues pushing at intervals below 10 seconds, so I recently wrote a node script to reduce the number of pushes necessary by comparing the base64-encoded version of a new render against the previous one. I document that here if you’re interested: Strategies for Self-Hosted Apps? - #5 by CyberMonk
And the Tidbyt folks may have something to say here, but I doubt screen longevity is a real concern, at least not when it comes to frequency of visual updates.