One possibility would be to run the battery down to 50% or so, then charge direct DC during the week, do a grid balance over the weekend.
A
custom charge controller could charge the battery safely and terminate the charge at say 4.1 volts per cell. (I say safely.. I don't know how the battery would respond to intermittent charging, but I assume this would be fine)
1 bar on a 2013 = roughly 800 Wh, 8 Ah. Two bars (for round trip) = 16 Ah (~1.6 kWh).
If you get 4 hours from solar over the entire day, then you need about 4A @ say 110V .. about 2x what you have now.
Powerfilm has a number of different thin film solar panels. Their
60W panel is 15.4V 3.6A, weighs about 3.2 pounds.
8 panels in series would give 123V (open circuit?), which is in the right ballpark to charge the 2013 battery. Total weight would be 25 pounds of thin film photovoltaics, call it 30 pounds with cables and electronics.
Total area required would be around 8' x 16'. And you'd have $6k in thin film PV just laying out on the ground. Perhaps not super-practical..