What is the safest and most convenient way to transfer between these two airports? Just plain taxi? Uber?

I am planning a trip to Europe leaving out of JFK (I am chasing after the LH747-400). Though I historically almost always flew DL (which flies to JFK), I thought I thought I would fly UA instead (which does not fly to JFK). I thought to myself that I might as well get all of my PQPs/ PQFs for the year.

I'll have a backpack, roll aboard and at least one checked luggage in tow. Apart from taxi and Uber/ Lyft, is there a safe and convenient transfer service between these two airports? I think Lyft would cost $100~120 each way.