That's what I call determination in putting together an award routing.

If I were you, about 5 days before the trip, I'd see if LAX-FRA on LH has become available. It will cost you $100 to change, but it would be worth it to reduce the number of segments and eliminate the taxi ride from LGA-JFK. Since you're beginning travel in a gateway city, you should have a good chance at a better schedule.

If you really want SQ from JFK-FRA, apparently UA ps service is sometimes available close to flight dates.