I have the same macbook with same battery problem and service center is quoting 13k Where are you getting quote for 4-5k?
Don't go to Apple service center. They will loot you unnecessarily. Apple itself uses 3rd party batteries so there is no difference if you get it fixed from outside.
I got the battery replaced for 3.4k, and now getting offers on olx upto 30-32k. I think its decent price.