Solar fence chargers are kind of a waste of money in most circumstances - you have to replace the battery every few years anyhow, so why not just get a battery-operated charger or a plug-in. Commercial solar-powered fence start at like upwards of $200, although if you happen to have an inside way to get an affordable solar panel and, what's it called, charge regulator?, then an astute person can sometimes do it for a bit less).
If you *did* have a solar fence charger, I would think it wouldn't be too hard to tap off the battery to run a light although it might necessitate housing the battery outside of the charger compartment where it's supposed to go (so you'd have access for another wire). I don't know what commercially-available timer could be used on a DC system like this, though (the only ones I've ever seen are meant for AC wall current) and you would have to have a timer to do what you want.
I don't suppose you could just buy a couple of the solar 'patio light' type things and install them with the panel sticking outside and the light inside the coop? i mean, 4x6 is a pretty small space, and if it were painted white over the entire inside I think you would have a good shot, at least, at a few hours' worth of bright-enough light from a couple of those patio lights.
It would be simpler in the long run, though, just to run electricity properly out to the coop, if you can manage it at all. This will be especially valuable if you live in a climate where it freezes in winter, so you can run a heated waterer or waterer base.
JMO,
Pat