That's not what Imperator meant in republican context, and that's not the title he regularly used. Imperator meant someone who is invested with imperium, and thats literally it, varying from a governor, to a general, to a high level magistrate, etc. It's true Egypt had one after Imperial integration, but your supposition as to why is a bit erroneous. It was used because literally anyone in control of a province had to have Imperium, and anyone with Imperium was Imperator. Imperator took on a different usage throughout the empire especially after the crisis of the 3rd century, and that's the one you're using. Augustus generally used the titles of "Princeps", "Pater Patriae", or "Caesar" as even by his later life it had taken on a political sense, not just a reverential one.

This highlights something wierd to me: why Baetica began depopulation far earlier than most of the Empire. It seems odd given it was not an at risk province, and wasn't valueless for trade, production, taxes, or recruitment until much later.

It seems odd for Britain to continue to prosper while Baetica didn't......

Well, the early "Byzantine" era (It's just late antiquity Roman, Eastern Roman if you must) did still have Latin usage for much of it until Heraclius at an official state level. So I couldn't quite go that far as I didn't catch the exact date. Maybe you're right though, if it's post Heraclius than it would be Greek Romans.

It would still have been referred to most often as Heracles, as Greek was the dominant language of the eastern portions of the empire. Latin never took hold as the lingua franca of the area, so along it Heracles is more accurate to what locals (and later Romans themselves in the area) would have called it.

But I digress, it is mostly a matter of which perspective you find most personally valid.