is there a cheap replacement battery for a 2016 MacBook?

you really shouldn't "cheap out" when it comes to a MacBook battery. that said, you can go to OWC’s website to shop for a battery for your specific Mac. they are known for providing quality Mac compatible hardware. and they also have DIY videos on their site so you can determine for yourself if it's an operation you are capable of doing, or if you will need to farm it out to a professional.

you must have misread the OP's question. they inquired about a new battery instead of buying a four figure new MacBook.

I have never seen a 4-figure battery for any kind of Mac.

Apple has a service where they install new battery in that model Mac, warrant the entire repair for 90 days, and it costs under about US$250.