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This can't really be automated at this point.
But one thing you may want to try is closing all EyeTV video windows (Remote Buddy provides an action for that which can you can add to the menu (if it's not already there by default)) - not the entire EyeTV application. On older Mac minis, graphics memory is a very scarce resource - and this should free up some of these resources, possibly already fixing the performance issues you see when using both applications at the same time.
Another idea for you may be to create a custom action in EyeTV to switch from EyeTV to Kodi. It could be composed of the actions "EyeTV > Close all playback windows" and "Behaviours > Activate Kodi Behaviour". Alternatively, you could also use "Behaviours > Quit currently controlled application" and "Behaviours > Activate Kodi Behaviour". You could then put an entry for that action in the menu or assign it to a button of your choice.