Note: This has no impact on your argument. I only mention it because I love me some physics.
Here's a citation, if you're curious:
https://physics.stackexchange.com/questions/34421/does-the-m...
The short version is that it will have less mass when the battery is discharged. However, you're not going to be able to detect a difference on your bathroom scale.
Again, this is being posted as a mere factoid and not meant to argue your point.
The other problem is generating the energy. Yes solar panels are a good energy source in an apocalyptic scenario, but the power they give out is really limited compared to fuel. 10m2 of solar cells will generate around 5kW per day (assuming no nuclear winter). With current EVs that would give you a range of around 10-15 miles once charging loses are taken into account.
https://groups.google.com/forum/#!topic/comp.society.privacy...
As impractical as it seems at first glance, the best post-apocalypse vehicles (barring an EMP scenario) would be electric since scavenged solar panels could provide the energy source for generations without significant degradation. Scrounging for electricity-based energy generation is much more practical than petrol-based resources that degrade, need refining, etc.
Batteries won't last as long as the solar panels, but as a personal observation, I've noticed that my parents hybrid Prius still has over 95% battery health after almost 15 years. High-quality batteries with well-engineered charge controllers have really impressive life spans in terms of maintaining their charge capacity!