We are end users like you so you need to ask Apple. It may be worth it.
However, this sentence is a strong clue to a possible non-battery cause of the problem:
"When working on photos or anything at length, the battery gets so hot I have to shut it off. I've literally burned my lap before."
You may have a computer placement problem instead. Apple calls these notebook computers, not laptops for a reason. Using it in your lap can cause overheating.
Without a rigid lap board, your clothing can block air intake and exhaust functions, resulting in high internal temps. Your air intakes are on the right and left sides between the bottom cover and the chassis. Air is exhausted from the area around the hinge in the back. Obstructing either can result in temps high enough to cause your computer to shut down to save itself from damage.
If you have any doubt about the battery, by all means have Apple examine it regardless of the serial number thing, However, first use the computer on an lapboard or desk to see if the heating recurs. Lap boards need not be exotic; for years I used a small plastic serving tray that my wife no longer used.
Here are the intake and exhaust areas that must be respected: