I think you are fine to light it and do a fuge back there! My initial thought was that it was too small but then I realized that the tank isn't that big either. The trick would be NOT to light it from the top. It looks like you can light it from the back and light it all the way down... I would do it. Your chaeto doesn't need to tumble. To do so is a newer trend. It does help keep detritus out of the chaeto -but more than anything it looks cool on YouTube and so it has caught on. I have never seen growth difference between tumbling and not. I can say that I have (sample size of one here) seen much smaller pod populations in the tumbling version. As for the detritus -I just pull my chaeto out, remove about 1/3 of it, and shake the rest in a container of water and put it back in the fuge. Harvesting and cleaning done at the same time. I would only rinse it every 2-4 weeks.
While you would definitely see a benefit in a larger fuge I think the lights down the back would give you ample "growth space" for such a small tank. I think LED strip lights or a flat LED grow light pushed up against the side would work just fine. Also in my past nanos the night to day pH swing was heavily stabilized by a fuge with a reverse photo period.
Unless you are running a super minimalist scape you probably have enough rock and sand to support more than enough bacteria for filtration. If you have gone bare bottom you are much better off scraping the fuge and adding more filtration to the back as suggested above. Been running nanos since 2002 and have done some pretty small but pretty helpful refugiums in that time. You DO need to give them enough light or they are in fact useless -but I think you can do that given the transparency of your "sump."
Figured I would at least give you a counter opinion.