Frequent flyer here. Been in and through NY lots of times. You say 3 hour layover, but if your flight is delayed for any amount of time, which happens frequently, then it is less than 3 hours. You also have to consider your time it takes to Deplane the aircraft and get outside to catch your ride. If your in back of the plane that could take 20 minutes or more just to get off the plane. 11 miles from JFK to LGA could take longer than an hour depending on traffic. Also security lines at LGA are always very very long. Could be up to an hour wait. Also once through security, you still have to get to your gate. The way the terminal is laid out it could be another 30-45 minute walk to your gate, depending on how physically fit you are. The terminal inside does not have people movers or a tram, so it’s a very tiring walk. There is one shuttle that you can board outside that takes you to select locations at different terminals, but if you’ve never been to LGA you will never find it, as it’s not clearly marked. I would recommend trying to change your flights.
IF you still decide to try switching airports at your layover then definitely Uber as it’s more reliable and there are plenty waiting at the airport. Uber also has vans and suvs to pick from. Make sure you are set up with the Uber app on your phone and have a payment method installed. Get yourself familiar with how to use Uber beforehand, don’t try to learn Uber at JFK. Worse case if you can’t get a van or SUV, order 2 Ubers and split up, which could also cause problems. If you do get 2 Ubers, then 2 of you will need the Uber app to order separately. Sign yourself up for clear pass before you leave. This can be done at your departure or departure layover airport. There are Clear Pass stations right before entering security checkpoints. Clear pass will get you head of the line privileges through security and will save you time at LGA. Don’t try to enroll in clear pass at LGA, have this done before you get there. TSA precheck is the same thing, but I am pretty sure you don’t have that. Lastly eat your Wheaties that morning for your OJ Simpson trek across LGA to your gate.