They can do a moveable lip that thing is no problem, however the real advantage for the DSI was its simplicity, there is not other advantage, it is simply simplicity, it diverts the boundary layer without the need of a splitter, on the F-35 it was added because it simply reduces weight and maintainace and therefore costs.What I'm saying is that given how the DSI works, having an movable inlet lip is still probably a much simpler and lighter solution than say variable shock ramps. Why would a solution like this be necessary? Probably because part of the requirements for the plane is to have it move at speeds beyond what a fixed DSI would allow, or perhaps it has to do with how capable the engine is.
I don't know how much more or less such a small change as a moving lip will increase RCS, especially if it's properly managed. Finally, the F-22 has a fixed inlet design with diverters. Perhaps a DSI offers better control of RCS (by controlling the direction of radar reflection entering the inlet?)
Adding moveable parts will cancel out its main advantage.
Is it the best type of inlet? no it is not, it is useful under some conditions, the F-35 only gets to Mach 1.6 so it does really pay off a DSI, same is the JF-17; the T-50 does not use it and on the YF-23 they used another solution and this was the wing leading edge was part of the inlet lip, they drilled some holes to take the boundary layer and they create a really simple inlet, the wing leading edge hardly generates a noticeable boundary layer so that was a pretty much smart solution.