Check the voltage as you start, pre starting it should read high 11, low 12. Drop when starting abd then 13+ and settle at 12+. 10.5v I assume is the voltage when not running also. If the voltage does not move above static charge then the alternator is putting nothing out. The car does not care what battery it has.
Getting to the alternator is pretty easy and a rebuild kit is about $60 from Bosch (you can get a cheaper brush only kit, I don’t know if it’s worth cheaping out here, I’ve done it with crank sensors and was nothing but trouble), of throw a rebuilt one in. A rebuilt is usually your old one cleaned with a rebuild kit on so it will be about competency and wallet size at this point. If you are just buying batteries off the shelf I’m assuming you are looking for the fastest option, if so a rebuilt one may be the quick abd easy option with regards to availability.
https://www.pelicanparts.com/techarticles/Boxster_Tech/82-ELEC-Alternator/82-ELEC-Alternator.htm
Alternator parts number for 2008 base 986 99760302204 cross references with 99760302200 99760302202 AL0869N
A quick glance shows
https://www.maniacelectricmotors.com/alternator-rebuild-kit-for-porsch-911-carrera-0124525106-0124525107.html
But says aftermarket brush kit but does cross reference part numbers to
F00M145651
And a search with that part number brings up seller CSS on eBay who I have gotten many parts from.
I include numbers for people who may be searching later and the links may well be expired by then.