Update:
After one week they called me and told me that their indicators show water damage (not entirely sure how/when, possibly even just damp over the last 8 years) and, because they have no spare logic boards, they won't be able to perform any work on it as it would fail their tests. They said they can only do complete work on it, not a partial repair (such as battery replacement). Irony is, they gave me a laptop back with battery disconnected (I tested it in the store) and had to take it back to reconnect 🤦♂️🤦♂.
So, now back to square one. As I really don't need new top case/keyboard (guy who took my laptop in even commented how nice condition it is in), I will do the replacement myself. SLODA and NinjaBatt are available on Amazon, and both have mixed reviews (especially about them failing in under a year, due to old age/chemicals) - which was the main reason to go to Apple.