I didn’t read everything above, but I could reproduce the issue just by following the instructions in the initial post.
I would propose a fix as follow: when the iNat browser tab looses the focus (or when it gets back the focus), if the current focus is there then the focus has to be removed from there:
Removing the focus from there should prevent the list of suggested taxa to drop down again (with the risk to click on it at the same time (this bug)).