Use of business class lounges depends on cabin of departure flight, and also OW status. If you use CX J to Taipei, then of course you can use the Pier Lounge business class section.
But, I suggest you instead save your miles and fly JFK-Tokyo-Taipei (TSA or TPE) on JAL on one Alaska award.
Alternatively, I suggest you fly JFK/EWR-HKG-Taipei (TPE) on Cathay on one Alaska award.
You could even stretch it and utilize the Cathay HKG-TPE-NRT flight, and actually book
JFK/EWR-HKG (transit HKG <24 hours)-TPE (layover in Taiwan of a few days/weeks/months) - NRT(destination)
all on Cathay on one Alaska award.
e.g.
CX 841 JFK 10:00 HKG 14:00 +1 --->CX 430 HKG 13:35 +2 TPE 15:35 +2 (23hrs 23 minute transit in HKG)
after a long layover of days/weeks/months
either CX 450 TPE 13:00 NRT 17:20 or CX 522 TPE 16:15 NRT 20:35

Keep hunting for the first class award space, especially JFK/EWR-HKG.

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If you insist on stopping in both Tokyo and Hong Kong ( >24 hours), then you will need separate awards, since you cannot mix two Alaska partners (JAL and Cathay).
BA Avios are very useful for this short flight (CX J to Taipei). But, you have to book early since BA blocks last minute Cathay award bookings.
There are actually 5 Cathay business lounges in HKG, so you can do a lounge tour! (I did it last February, a glass of Moet champagne in all business class lounges) The Wing, Pier, Bridge, Cabin, and the new Deck, plus the Qantas lounge (nothing positive about the Qantas lounge)