The question is more : can it be done with the tech they have ?
Well, the layman in me wants to say no, that is FAR, FAR away from what was ever done before and usually avoided at all costs.
But, the systems engineer in me says: Its just a few constraints more, some more requirements in certain situations and nothing that suggests that these requirements contradict each other or might do so in further iterations. Hell, even the fuel combination used has so many favorable properties to solve the problem, that it should be no showstopper.
It MUST be possible with current tech.
Also, I don't think igniting the engines is the problem, because they seem to have worked it out with brute force. A more elegant solution might be including closed-loop control over the engine already early during ignition, instead of the transition to main stage - but that requires flight test data. Brute force is fine for that now.
The main problem they have IMHO is shutting those engines off in a safe way. Its way easier with gas generator cycle engines, with full flow staged combustion, you need to stop the full flow - which can't be done suddenly without killing the engine. I think this could end in a redesign of the Raptor engine. They need to keep the engine cooled without venting methane - so one half of the engine would need to be capable of switching into a limited gas generator-cycle operation without venting methane rich exhaust into the main combustion chamber or the world around the engines. Or additional pumps take over - for example, how painful would it be, to add a small gas generator and turbine to the fuel pump shaft, to permit pumping methane at reduced flow rates and/or pressure until the engine has safely reached the parameters that can permit a full cutoff? This wouldn't need to be very heavy or long-lasting, since we talk about <10 seconds until the engine is really cold (and not a fire-hazard until ignition).
Maybe such a solution is heavier, but simpler to implement as further reducing the minimum throttle settings of the gas generator.
Also, a method of inerting the chamber after each shutdown is needed. Without, moisture could freeze into ice INSIDE the engine and make reignition more dangerous.