To help you more...
The key-fob itself uses those (
Gosh Darn) Lithium coin batteries.
In mine they use 1 2032 - which is kinda' important due to several factors...
One - is their size...
But more important is their rated voltage at that size - for a given draw of current in mA
CR2032 can supply up to 200mA draw for a rated length of time RATED for 3 V
CR2025 can supply up to 165mA draw for a rated length of time RATED for 3 V
CR2016 can supply up to 90mA draw for a Rated length of time RATED for 3 V
So since you have a key that emits an RF signal - the 3V supply has to produce the ID and any other string of information in a manner the receiver can decode it using a receiver designed for being FM/PWM or AM or even DTMF on FM.
No matter how you look at it, if the Battery cannot supply the right amount of current - the system can trigger a fail and not start your car.
It's not a bad idea to recheck the supplier of your battery and just get another and try that one to see if it can work - I'd hate to know that this could have been fixed because the battery off the shelf was also dead or past-dated and expired - you just needed to clean and re-seat the contacts and use a new fresh battery to get your car back on the road.
The bad thing would be that the
Cars Battery was good, it was the
Keys Battery that was causing the fault.
So if you put in the correct ones - great, but if one goes bad or is not at 3V - it may be having higher output than 3V or less than 3V (which can be just as bad) - can cause the signal to become distorted when its sent - then the Door lock would seem to work because the PATS senses the control signal - but the START control it needs is not being recognized.