Best web browser for battery life other than Safari?

This machine does Quality for macOS UpGrades all the way to Catalina and with a newer OS - the newer Safari.

Mojave is the last version of macOS still supporting 32 Bit Applications, drivers and extension wherein Catalina Stops All support for anything 32 bit .

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install. Make sure to use Only Safari to commence the download as Others may not work. Safari will open a redirect and open the Apple Store where you can commence the download. If an error message appears, that is probably because one is using another Browser.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Make a Time Machine Backup Before attempting. Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina & Big Sur support 64 bit and there is no workaround.

Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal operation of the OS and may even Corrupt the OS requiring a Reinstallation

Sense the battery is a major issue from your posting.

If this machine from 2012, still has the original battery as when new - a Nine Year Old battery is lucky if it holds a reasonable charger for any portable usages.

Side-stepping that obvious issue, attempting to get a newer Browser for Mavericks is IMHO - not lightly to solve the real issue - Battery.

Batteries are considered Consumable Products. Over time it will degrade to a point where it needs to be replaced. The usage is normal as the capacity will go up and down all the time and never stay always at 100%. Apple Batteries are rated for 1000 Full Battery Cycles and / or 80% Capacity before needing Evaluation or replacement.

It might be better to to have the battery replaced ( if required ) and then consider a move up to anything from anywhere from Yosemite all the way up to Catalina

Welcome and whatever direction is chosen - very best

Does a web browser really affect battery life.

And as you have pointed out you are almost OCD, no one can make allowances for that other than your self.