Could be I am wrong but to LAX I would book with LH 17.5k M&M J.
LAX-LIH with BA X & the return to NYC with UA X?
Didn't know this thread exists. I almost made one myself recently.
I don't know about booking Y, but for J, I think you're right.
NYC-LAX on UA booked with 17.5K LH/M&M.
LAX-LIH on AA booked with BA (not sure how much but J is just domestic J so it's not much of an upgrade over Y).
LIH-NYC doesn't exist, you'd have to fly to HNL first (costs/options for that trip can be found in the wiki of the Hawaii General Thread). Once there, HNL-EWR in J is cheapest booked with 30K SQ IINM.