Wow, so many ways to make something as simple as 4 gears difficult.
Your bike has vertical drop-outs, so form of chain tensioner is needed, might as well use a short cage road derailleur.
To shift between only 4 (or maybe 3 or 5 ) ratios why have any more than 1 shift mechanism?
If you want sleek and clean, why have a front derailleur and a chain tension device at the rear?
Really, a small freewheel and rear derailleur is going to be simpler, cleaner and offer superior shifting to all this other stuff.

Edit. It's going to be way cheaper too.