A dead key battery has no affect on its ability to start the car. EWS & FZV are complete independent systems...as well as the 3rd feature of the key DWA.
EWS - immobilizing system
FZV - central locking system
DWA - anti-theft alarm siren system
Did you check the 2nd time to see if the CAR'S battery was dead again? You can try to reinitialize your remote keys to lock/unlock & arm/disarm the car...but again if the problem is the car won't start...it's your car battery that's the problem, not the key's battery.
If possible I would leave at least one door unlocked until you get the driver's handle fixed. It's the only MANUAL way into your vehicle when there's no electrical power.