This is fairly typical for lead acid batteries and not an indication of a problem.
If you've ever ran down the battery trying to start a troublesome engine (to the point it won't turn over), you will often find that if you let it sit for a few minutes, it will be able to turn over the engine.
As far as sizing cables, look up a voltage drop table.
- 2000w draw at 12v will be drawing about 170amps. Then take the length of the cables from the battery to the inverter.
- The solar panels at 500w can be done similarly.
Then you just have to decide how much voltage drop you are willing to accept. If the cables don't get crazy thick, I would shoot for 3% voltage drop or less but 5% would typically be acceptable.
Ideally, having a short distance from batteries to inverter is helpful. The solar panels are more difficult as they are likely on the roof but even there, don't bring the cables down at the rear of the unit if the battery bank is at the front as it will result in a lot of length which increases voltage drop.