There is no phase lag of the sort you allude to. (realclimate.org)
Actually, I think something like this does happen to a modest degree in El Nino, superposed on the natural radiative damping to space, but big enough to give you phase lags a la Spencer and Braswell. (gfdl.noaa.gov)
The feedback is mentioned or assumed in many of the references cited above, e.g., see quote by Genthon; Lorius et al. (1990) remarked that Greenland drilling then underway «should allow a better determination of the relative timing (phase lag) of climate and greenhouse forcing» (p. 145), but the wider community had thought little about a lag. (history.aip.org)