Testing my model using Hansen's linear response times (which leaves me with only one variable to play with, climate sensitivity) I need to use a climate sensitivity somewhere between five and 5.5 to get a match with the observed temperature trend between 1970 and 2010.
This could indicate that my model is wrong in other respects - I will need to read Hansen's paper carefully to check this, as he does mention a long-term sensitivity of around six degrees.
I should also note here that I am aware that my model is attributing all of the temperature increase between 1970 and 2010 to CO2, making the assumption that other forcings cancel out over that period. This is likely true for things like solar forcings, ENSO and so forth. However, aerosols are still an issue.