Based on my experience, save your time calling the EXP desk and do the searches yourself using EF (buy your friend a pint then invite yourself round to his to use his computer perhaps?). You can search based on route, for 7 days at a time. And yes, you'll be looking for C inventory. Find flights that are C2 or greater (and presumably operated with a 77W, which is the newer 777-300ER). With that you can then call and buy the ticket on the phone directly from the EXP agent, or you can go online and put a hold on the itinerary you want and finish the booking on the phone; it's better not to buy it outright first as they cannot apply the SWU until the itinerary is ticketed, and during that time your inventory could disappear. By having them apply the SWU to a held reservation (or having them take care of the entire matter), you avoid that potential problem. Good luck!

I imagine you'll find it rather difficult to clear 3 people into business as a platinum. However, this is the late flight (which typically is much easier to upgrade due to the relative lack of onward connections from JFK) and it's on the old equipment, so you have those things going for you. One thing I would say is that you should prepare yourselves that you very likely won't be seated together initially. Also, remember that you'll get hit for extra APD if your upgrade clears (assuming you're not connecting in the UK from an international flight originating in another country), which will be some US$300+. Also worth keeping in mind is that Platinums using SWUs are ranked below all platinums and golds using miles+copay on the PALL list; this is because there is no VIP2 or VIP3 codes. Also, your ranking on the PALL list will be determined by check-in time, so checking in as early as possible would be very important (though if business is full, due to the above you're very unlikely to get the upgrade for 3 people anyways). Good luck!