Are Macbook Pro batteries interchangeable?
They are generally in the same generation. I'm not sure that far apart.
Here on iFixit, they appear similar, but I wouldn't be sure if the connector is exactly the same. It might be worth a shot?
Best of luck.
No the MacBook Pro different generations did not have the same battery so you cannot swap them. Some models, in particular the 2010 and 2011 models, did share some components, but you need to compare the part numbers and the EEE codes.
As for your issue, the cycle count for a battery in the 2009 MacBook Pro is 1000, not 300. And even that isn't a hard number; it's just that the battery is expected to maintain 80% functionality for 1000 charge cycles or 5 years. That was introduced with the 2009 models, not the 2010 models.