Is the following system LTI (finite speed of signal propagation)?
No, if your system can (with sufficient accuracy) be modeled as a linear time-invariant system, then it is fully characterized by its impulse response. Since the frequency response is the Fourier transform of the impulse response, the system is also fully characterized by its frequency response. The frequency response does not only describe steady-state behavior but also transients (because it simply is a full description of the system). See also this answer to a related question.