Yes I would always code in a new battery, this tells the charging system you have a new battery and therefore charges (over time) differently.
With regards to the serial number all that is important is that it is one character different to the current one so just add 1 to the number you allready have and it will be fine.
As per my previous post I dont know this issue but suspect that once you have coded and tried to clear the errors they will come back as there is a wiring
problem and a new battery wont
help, but hopefully somebody who has dealt with this issue before can confirm.