Coming to the battery performance, the 3,050 mAh battery unit is quite robust but is capable of only 400 shots compared to the 700 that you can get on the Sony A7 III. You can, however, engage the power saving mode in case of normal mode, and get double the shots on the Lumix S1. Sounds a bit of a shot in the dark, but worth the try.
All in all, the Panasonic Lumix S1 is a complete package and possibly the best mirrorless camera in the market today. Yes, the Sony A7 III does deliver a few potent hits and even costs lesser than the Lumix S1, but you cannot take away the fact that the robust build, weatherproof sealing, complete port and slot combo, and the L mount alliance, all form a package that brands are going to find very hard to beat. So, if you are looking to buy a mirrorless camera, and can afford to shell out a little more than $2000, you should definitely go for the Panasonic Lumix S1.