Going supersonic is, as Engines says, one of the problems of a four-poster Pegasus layout. PCB (afterburning on the front nozzles) was proposed in the early 1960s for the P1154, but was needed for vertical landing - which was probably a non-starter because of hot gas ingestion.
The shaft-driven fan was a great idea in theory but has fallen down a bit on execution and weight control. Maybe lift-plus-lift/cruise would have worked out better, but it is hard to say.
Fundamentally, stealth+STOVL+supersonic is incredibly difficult: STOVL+supersonic, or stealth+supersonic, are tough enough as it is.
Unfortunately, the decision to attempt that solution - further complicated by CV and CTOL versions - was based on very optimistic assessments of performance (driven by weight) and cost (development, production and operation).
The OEW of the F-35B is one-third higher than was hoped for at the start of the program (1996), and we all know about the costs. If that had been known, I don't think the program would have started.