First you don't want to furl flat. You want it to be thicker in the approximate middle and tapper out as you go towards the ends. The sails may look flat but they are constructed with a belly that bows out in the middle. Sail manufacturers can put foam or rope on the leading edge of the sail. It's not thick enough to affect sail performance when the sail is fully out but will improve the shape when partially furled. But even with that furling in 70% of the sail will not perform that well.
The proper approach is like you suggested, get a smaller sail. Stu Jackson over at the 34 OA sails in San Francisco. He has two head sails. A winter season sail when the winds are stronger and a summer season sail.
We sail in the Caribbean. So we had the same issue as you, poor sail performance because of too large of a headsail. So we worked with Precision Sails to have a new headsail made. We ended up with a 115 with a Yankee cut and a 10-inch raised foot. We have had the sail since April and have about 700 nm under the sail. We love the increased visibility and the performance is much better.
We purchased during their Black Friday sail. It was about 20% off their price.
Good luck and fair winds,
Jesse