I know we can write a trigger on UserServicePresence object
According to this, you actually can't.
Apex triggers aren’t supported with UserServicePresence.
Sharing rules aren’t supported with UserServicePresence even if the
OwnerId field is enabled.
In API version 41.0 or later, UserServicePresence records can be
deleted programmatically. The “Customize Application” permission is
That being said, if you can access those objects via apex, I believe you could try to build a custom console component that could insert/update those records when the console is accessed:
Custom Console Components
The only thing I'm unsure of is whether or not the components can be developed to execute automatically or if users have to click on them to activate them.
Otherwise, Maybe a custom visualforce tab that is the default tab for the app that interacts with the records on page load could also work. My concern here would be that the console likes to preserve the user's state which means it may not automatically direct them to the page the first time they log in.
If you're in lightning you can try a lightning component but I'm not as familiar with adding those into the console app per-say. I couldn't find documentation to suggest there is a way to do this so you may still end up needing a VF page. Alternatively, you can develop the component in a way that it can be placed onto any arbitrary detail record page and add it to all the records your reps might view but you will need to be careful while developing it as it could flip the status mid chat which you may not want (make sure it only goes from offline to online for instance).
You'll have to play around to see if any of this will work for you.