Will be a pretty much insurmountable challenge getting four J award seats on one flight on a specific date.

Regarding Amex miles, your first decision is which program to transfer them to. Aeroplan is a good choice, IMHO. From there, you can explore multiple *A routing options to include LAX-ORD/JFK-IST on TK, LAX-FRA-IST on LH, even LAX-LHR-IST using a combo of UA and TK, etc. You may have to split your party into two in order to get the class of service you want for your entire party.