It does not; there is no "speed of light limitation" in physics, per se. There is only a requirement that any two observers in different Lorentz frames observe causality in the same way. The state vector of entangled particles is one state vector. A measurement is the application of the operator corresponding to the entangled observable which requires the state vector to collapse to an eigenstate, but the observer who has not done the measurement has no idea what the eigenstate is. Therefore, no information has been transmitted from the location where the measurement was done to the correlated location, and the observer at the correlated location must still do a measurement, even if the state vector is now "known" to be an an eigenstate by the original measurer. Because no information has been transmitted, no causality violations can occur.
but the solution, I suspect, rests with higher dimensional dynamics. In other words, the photon is not moving faster than the speed of light in 4D.
Relativity is already four-dimensional, so you must mean five-D. But if the measurement produced an eigenstate of its operator in the entangled state vector by way of a higher dimension which was space-like or time-like, causality violations would be possible in 5-space. The hidden dimension [if that is your explanation for the violations of local realism present in EPR] would give rise to observers who were always inside each other's light cone in Minkowski space [4-D] but who could be in opposite domains of causality in 5-D.
Not outlawed, but it would be strange.
And you would have to have an explanation for why fundamental interactions [gravity, say] which ordinarily have dynamical effects do not propagate through a time/space-like dimension...
And yes, I do mean five dimensional physics as the most promising explanation for phenomena such as EPR (in my view.) For more, please see P.S. Wesson's Quantum Mechanical Consequences of Five-Dimensional Relativity
Roger Penrose also observed that a simultaneous measurement of a spatially separated entangled photon, would result in an unacceptable paradox where the measurement of each determines the other based on our current physics. He calls for a new kind of physics, but again I suspect the solution rests with a higher dimensional dynamic rather than a 4D or compactified extra dimensions (Kaluza/Klein.)