Why limit yourself to LAX-JFK flights if all you want is a lie-flat? DL SFO-JFK and UA SFO-EWR flights are on lie-flat equipment and these fares are significantly cheaper than LAX-JFK/EWR fares (it's also slightly less miles than connecting through LAX). They both start at $1299 one-way for the time period you are looking which have a 14 day advance purchase requirement. The cheapest fares on these routes are $1199, but they have a 21 day advance purchase requirement and you are too close in to book these.

Delta does not have any D1 fare filings on SEA-JFK which means the only way to price out connections through SFO or LAX on one ticket is with multi-city and separate fares on each hop. Further, they will require a stopover (connection over 4 hours) in order to book on one ticket. You will have to book fares on separate tickets if you want a connection shorter than 4 hours. This connection just squeaks in at over 4 hours and is bookable on one ticket. The price on the SEA-SFO FC fare below is $243 and the SFO-JFK fare is $1299 for $1542 total.