I finally gave up trying to book this mixed airline itinerary. I found another all LH itinerary that has decent connection and was able to book it on LH website and get the same price showed on Matrix website. 3 of the 4 legs are on LH so I was also able to get advanced seat assignment. According to the LH website I will only be able to earn 50% mile on 2 oversea (long flight) legs. Will earn 750 mile on flight from Munich to Rome. There is no mile earn showed for flight from Athens to Munich on Aegean airline.

Now I just need to decide which FF program to sign up to get the most benefit from this trip. This ticket is for my relatives who don't fly often. They only fly from LAX to Japan or China once every year or two.

Here is the itinerary that I finally booked for their trip

Los Angeles (LAX) to Rome (FCO) - Sun, Jun 15
United Airlines, Inc. Los Angeles (LAX) to Munich (MUC) - Sun, Jun 15
United Airlines, Inc. 8861 Dep: 5:20PM Arr: 1:40PM 11h 20m Airbus A340 Economy (V)
OPERATED BY DEUTSCHE LUFTHANSA AG Layover in MUC Mon, Jun 16 2h 20m

United Airlines, Inc. Munich (MUC) to Rome (FCO) - Mon, Jun 16
United Airlines, Inc. 8874 Dep: 4:00PM Arr: 5:30PM 1h 30m Airbus A320 Economy (V)
OPERATED BY DEUTSCHE LUFTHANSA AG

Athens (ATH) to Los Angeles (LAX) - Sun, Jul 6
Deutsche Lufthansa AG Athens (ATH) to Munich (MUC) - Sun, Jul 6
Deutsche Lufthansa AG 5917 Dep: 8:35AM Arr: 10:15AM 2h 40m Airbus A320 Economy (Q)
OPERATED BY AEGEAN AIRLINES S.A. Layover in MUC 2h 0m

Deutsche Lufthansa AG Munich (MUC) to Los Angeles (LAX) - Sun, Jul 6
Deutsche Lufthansa AG 452 Dep: 12:15PM Arr: 3:30PM 12h 15m Airbus A340 Economy (Q)

Cost per passenger (including taxes & fees) $2,052.20
Total cost for 2 passengers $4,104.40