Does anyone know how I can make this Macro run in the background? meaning I don't want the app to move to the front? Note, it does have an important move and click mouse action.
All told I doubt you can achieve this. The approach you've tried is interacting with what's on screen, so can't act on Tweetbot if it's not frontmost.
Keyboard Maestro can send a Type a Keystroke Command to some applications in the background. Like so, sending the Refresh shortcut to Tweetbot:
Unfortunately, not all applications can receive a background keystroke, and that's the case with Tweetbot v2.5.8. If you're running a newer version of Tweetbot and want to see if it behaves differently, duplicate the shown macro which is just the Type A Keystroke action, and use the Gear menu to select Send To > Tweetbot.
The only other approach available that I know of brings Tweetbot to the foreground for a split second so probably doesn't help.
Ultimately, in case you don't know, Tweetbot has a 1–2 minute lag between refreshing itself. This is a limitation Twitter put on third-party apps sometime in 2018.
Right, even my Macro doesn't do exactly what I want. it refreshes (I think). But it doesn't scroll to the top of the timeline which is why I can't just use the ⌘⇧ + R keyboard shortcut.
To get to the top of the timeline, after refreshing you can use the Type A Keystroke action to press the Home key which, if you're using a notebook, would be FN+Left Arrow.