How can I automate clicking in a Chrome extension popup with a hotkey?
I haven't found a free solution, but I've been able to solve my problem with the free-trial version of Keyboard Maestro.
KBM allows you to record key clicks and mouse motions, and with an option on the Recording process, you can include real-time time delays that can then be edited to speed up the process without it zipping along too fast for the apps and app switcher to keep up.
On thing about KBM that is very useful, as hinted at in my problem description, is that the mouse clicks are initially recorded relative to the upper right corner of the screen or the frontmost window, BUT, after recording, you can change what the zero-point is for the mouse coordinates. In my case, changing the mouse clicks to be relative to the upper right corner means that they should be pretty stable w.r.t. where the RenameTabs Chrome extension dropdown dialog appears.
I still don't know how to capture the current window size, which would be required for a totally free solution. One option I've thought of is to use another Chrome extension, Easy Window Resize, to ensure that that the window is always at a predefined size so that I can indeed use Cliclick relative to the upper left corner. That could be a free solution.
I probably won't go that way for my own purposes because, the other features of KBM make it look like a worthwhile investment for me.
DISCLAIMER: I am just a customer, currently using the free version of KBM and am not affiliated with the company in any way. I haven't even paid them anything yet.