Actually the idler is the problem that's being solved here. It's noisy and inefficient and it creates other problems I'd rather not get into.

The solution is to use a small chainring, which in turn required a custom wheel with an internal hub and a custom sprocket and a custom chain. Although I painstakingly measured clearances ahead of time, somehow in the end things didn't line up (I think because things flex under load).

So now I've discovered I need a new frame (although it will take a few more hours before I know if it will need to be a custom built frame or a stock frame). However, since I've already spent hundreds of dollars and hundreds of hours, I don't want to invest more time and money on the frame unless I'm sure I'll be happy with the result. Therefore I'm looking for a temporary solution to test out what I've done so far and for that I think extending the dropouts is my best bet.

And that is why I made this thread to inquire about how I can extend the dropouts. So far all your suggestions are steps in the wrong direction.