The check engine light would most likely be related to an emission control device or sensor. The alternator light would indicate a problem with s separate component in the charging circuit.
Looks like 2 problems for the price of one! Opinion only: I do believe that there aren't any 132 month batteries out there. Might simply need to change your automotive battery and re-evaluate the situation.
It would be cheaper than the drive to the dealer, lost productivity, an hour or more of labor plus the cost of the new battery.
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