Those solar chargers are trickle chargers, and would never be able to recharge the battery at or near the same rate as the discharge from the battery charger. The length of time the lead acid battery would last depends on many factors-Charge rate of your battery charger, battery pack capacity you're charging, C rate at which you're charging at,amperage rating of the car battery being used, outside temperature, etc.
Then, you would have to recharge that lead acid battery at home, either with a wall charger, or some sort of solar trickle charge setup, but still involves a lot of hassle either way you look at it.
Ihave a Triton EQ charger that I use at the field. It has a max charge rate of 5 amps. Iused to use the battery in my truck to charge, but now use the generator. Isafely charged packs up to 2650ma at near 2C, probably 10-12 packs in a day of flying, and never even noticed a slow crank afterwards. But, keep in mind, if it's not a deep cycle battery, a regular car battery is not designed to be discharged and recharged over and over like that. It's designed to have reserve power to initially start the vehicle, then the Alternator/Generator takes over.
If your pack is 10k ma 5S, even at a 1C charge rate, you're still talking about a 10+ amp charge rate. You can drain your car battery in just a few hours at that rate.
My little generator runs for 5 1/2 hours on 1 gallon of fuel, and it only cost me $130.00. Not a bad option if you really think about it.