Your comments

Sorry, but I don't understand the question. Do you want to change the update checking interval?

Yes, I'm familiar with the issue (which is different from the one presented in the original suggestion).
I'll see what I can do about it.

I believe the issue has nothing to do with the tweaker.

If you think otherwise, e.g. the issue occurs for you only when the tweaker is running, please let me know.

Closing due to lack of input.
Please reply if you have more information about the issue.
Taskbar Inspector always opens near your cursor, I think that's most convenient.
Why would you want TI to appear at the top when you click at the bottom?
7TT doesn't make permanent changes to the system, so if you disabled autorun, restarted, and the issue still wasn't solved, it means 7TT is not guilty here.
Uninstalling it won't change much.