Up until recently, perhaps, one might be able to access club/lounges at JFK with a valid same day BP, even if flight is departing out of a different terminal. But, it's no longer the case lately at JFK's T1 (KAL Lounge) for Priority Pass member without BP for a flight departing out of T1, as TSA might simply not allow passage. I can't say for sure with the Priority Pass/Air France Lounge at T1 by just announcing that one is going to the lounge & waive just another BP - and to deal with getting thru security screening twice, once at the terminal for the Lounge, and a 2nd. time at T5 for JetBlue's departing flight.
OP is flying JetBlue out of T5, and it would be a lot easier to just dropoff the bags & complete the check-in, and get thru security (even with Pre-Check, the lines can be silly long & slow during the morning rush - took us 45 minutes to get thru recently, as others ahead of us were creating bottlenecks. ) - otherwise, s/he would be spend time going thru security just to get into T4 or T1, airside - assuming lounge entry is allowed, regardless of whether reception will be checking for a valid BP, and, then to exit and head over to T5 (assuming that s/he already spend time checking in & did the checked luggage dropped off) and clear security - before reaching airside, allowing for time in the security line, and then, hangout near the boarding gate.
Our JFK layout is not quite the same or that familiar to someone used to flying in/out of TPE or HKG - with centralized terminal security & check-in, friendly layout for lounge access airside, etc.
https://www.prioritypass.com/en/Loun...-International
http://wingtipslounge.com/entry
Besides, OP would've likely been in the air for maybe 20+ hours with 1 or 2 connections, and could use a refreshing shower during the layover, available in the Airspace Lounge, airside at T5.