There are several issues involved here.
The recall program is only valid for the original battery which shipped with the laptop from the factory...so that no longer applies since Apple replaced the original battery for you.
This laptop is now considered "Vintage" by Apple which means repairs will only be performed as long as the necessary part is still available. In this case, the necessary part is the Top Case Assembly which includes the Keyboard, Trackpad, and the glued in Battery. This part definitely won't be available for long since a keyboard failure requires this part. A Trackpad failure requires this part. A Battery failure also requires this part. So you can see this part is needed for a lot of different failures so existing stock will be quickly depleted. Thank Apple for gluing the battery to the underside of the palmrest & trackpad so keeping a "battery" in stock is nearly impossible once Apple stops producing the Top Case Assembly. Apple does mention they will allow battery replacement for up to 10 years, but this has a very critical footnote "as long as the part is still available". We've had better luck with Apple supplying batteries for the "Vintage" MacBook Airs, but those laptops have removable batteries, however, even with those laptops Apple refused repair multiple times when later they did allow for repairs.
I am not sure of the warranty on a replacement battery, but I think it only has a 90 day warranty (1 year at the most). Again, the small print comes into play...because it is only possible if the part is still available. Even if you were able to get a repair today while the laptop is "Vintage"....so if the part fails next week, you may not be able to get a warranty replacement if the part is no longer available due to the "Vintage" nature of the laptop.
At this point, you will need to get a third party battery replacement. Unfortunately the quality of third party Lithium Batteries is extremely poor even when purchased from a reputable vendor such as OWC or iFixIt. These vendors do suggest professional installation due to the difficulty of removing the battery and the risk of damaging other components (including the new battery). I've seen a lot of users post on these forums that their laptops had issues after they performed their own battery replacement because they usually unknowingly accidentally damaged something.
I just installed an iFixIt battery into a 2015 laptop and the first battery I received was defective (lost nearly 80% of its capacity in two days), but I easily received a warranty replacement which seems to be holding up fine so far. In the past I have had premature battery failures from OWC as well. It is just the nature of the third party Lithium Battery marketplace.