Can a lifting body work at supersonic speeds?
Lifting bodies were extensively used for reentry vehicles. In the Fifties and early Sixties there were two camps with different ideas how to go to space:


The Air Force wanted to increase vehicle speed and altitude incrementally such that finally a horizontally-launched, winged vehicle would reach space. Since the materials and propulsion were still lacking, they started at both ends of the flight envelope: One culminated in the X-15 research program, and the other researched how best to land a vehicle suitable for re-entry. This produced craft like the Northrop HL-10, which is shown below.




Northrop HL-10 with B-52 launcher overhead (picture source)


The US Army, on the other hand, tried to increase the size of rockets, traditionally used as an alternative to artillery, and got a helping hand from some other ex-Army guys who had lived through exactly the same split of technologies, where the Wehrmacht preferred rockets and the Luftwaffe preferred flying vehicles.


So not only can lifting bodies be made for Mach 3 and higher, they pretty much originated from the work done to fly at hypersonic speed, at least for a short while.