Here is how I would do it:
sv_cheats 1 and
host_timescale 0.1 so that time advnaces at 0.1x speed. This will increase the accuracy of your measurement by a factor of 10. (Maybe 10's a bit much, but it's what I would do since dividing the measurement by 10 is trivial.)
- Set up an enemy right in front of you.
- Start the clock and switch to a bullet-based weapon.
- Hold down the shoot button during the switch and stop the clock when the weapon does damage (not when you hear/see it fire, which might be slightly out of sync with the actual damage dealing).
Note that you don't need to fire a weapon to start the clock here; your starting point is when you press the "switch to another weapon" button. This is because you never have to wait for something before you can switch to something else, with the exception of Heavy primaries needing to spin down or hauling buildings.
Once you've done this a few times with regular weapons you can go from Degreaser to Shotgun to see if that measurement matches up with the expected value.
I won't help you with the sap-and-shoot measurement because Spies that do that are jerks. :P (Seriously though: Since there's no delay before you can switch regardless of the weapon, the measurement won't be different than anything else.)