There is absolutely nothing to do at either LGA or JFK. If you are insistent on staying at one airport or the other, I would hop in an Uber at LGA on arrival and head for your JFK hotel and spend the night and the next day there. Might as well get that ride out of the way and minimize any remote traffic issues. The JFK area is also minimally nicer than LGA.
But, understand that there is nothing to do at either place and unless you are a fan of hotel restaurant food when you have the opportunity to have a good time in NYC, you are passing up something fun.
Given LGA's proximity to Manhattan, I would absolutely head into Manhattan, spend the night there, have most of the day there and then head out to JFK the next day, either by Subway/LIRR or Uber (depending on your preference). Given that you are not in a hurry, just give yourself more time.