Override image template for field
It's possible but very difficult to do.

You would have to alter to output of the image formatter to inject a theme suggestion on the image_formatter template (or create your own image formatter).

Then you would need to inject the suggestion from the formatter to the image theme function (could be done in a preprocess).

The issue is that altering the output would probably be something you would handle in each unique case, so you would basically need to create your own image formatter.

It's not a lot of work, but a bit tricky.

TL:DR;
Yes it's possible but you don't want to do that.