How to fix a Macbook Pro (late 2013 model) with liquid damage?
I would say that you would need to go ahead and use the Genius Bar option to repair your computer.

My reasoning is that water damage can cause errors in strange ways that, upon first glance, seem to be straight forward, but are in fact impossible to properly diagnose without professional diagnostic equipment. For example, in your case, I doubt that the problem actually lies in the battery. Rather, the water damage may have caused a part somewhere along the battery-charge monitoring path to fail rather than the entire battery, but that is simply conjecture.

For that $800, a trained technician will take apart the entire Mac and inspect each piece for corrosion and/or liquid damage. Each will be repaired and replaced and then the computer re-assembled and tested and a warranty will be issued for the work. Typically 90 days or the remainder of the existing warranty if any. Local regulations might extend these as well. The only reason this doesn't cost much more than the new Mac is that the parts are not all scrapped and some have repair / reuse value to Apple and they can pass the savings on to you.

Definitely not what you wanted to hear, but I hope this helps.

Caveat: if you can pinpoint that the battery - and only the battery - is the issue without a shadow of doubt, replace it yourself; it will be much, much cheaper.