I don’t know if anyone has tried this route but you could Uber it or if you’re brave bus it to Woodside from LGA. Jump on the LIRR to Jamaica then the Airtrain to JFK. Airtrain is $5 each way and the LIRR is $7.25 off peak and $10.25 on peak. Not sure how much the Uber would be but it’s not a far ride at all. Probably would be faster then Uber between the 2 ports also.