A battery replacement from Apple should only be about $200 US (this is the mail-in repair price...Apple Stores and some AASPs utilize this service as can some users in certain regions like the US). If an Apple Authorized Service Provider orders the part from Apple and installs it in their own shop, then the cost may be a bit more, but it should not be anywhere near $600 US. I would expect a price closer to $300 (part & labor) since the cost of the part for the AASP is almost the cost of the mail-in battery repair. The $600 price tells me they are quoting you the price of the Top Case/Keyboard Assembly instead of the Top Case/Battery Assembly where the latter is much cheaper, but only available if the battery is below 80% of the original Design Capacity, or the Apple Diagnostics indicate a battery failure (or the battery shows a battery condition of "Service Recommend" within macOS). A swollen battery should qualify for the Top Case/Battery Assembly price...shame on the AASP for charging the other price. Try getting a repair quote from another AASP, or see if you can mail the laptop to Apple for the repair (only available in some regions).
Here is an Apple article which has the prices of battery repairs (assuming the mail-in repair option whether by the user, Apple Store, or AASP):
Mac Repair & Service - Apple Support
Edit: I should clarify the Top Case/Keyboard Assembly and the Top Case/Battery Assembly I mentioned is exactly the same part, but the latter is referring to the "Battery Only" price for that component. Apple allows an AASP to purchase (rather exchange) the part at a lower price as long as a battery failure is confirmed (even if the part is damaged). Some newer techs may not be aware of all the nuances of servicing Apple products. Yet another downside of integrating so many items into a single part. The Top Case Assembly includes the Palmrest area, Keyboard, Trackpad, Touchbar, Speakers, and Battery. Just hope you never have the speakers go bad or it will be $600.
