No, I can only record the time the light has been switched on.
Currently, you have the watchdog timer set to 2 sec and clearing clicks after this. As a practical matter, maybe using 1 sec for this and mode mem is a workable trade off?
If more than one timer is possible, or something to similar effect, you can also set a shorter timer to count the number of consecutive short ON periods (decrement clicks if longer) as a reasonable approximation of fast clicking. I can't think of non-pathological use cases that would violate this, especially with reverse clickies. I suppose in the worst case this also works if we're willing to give up mode memory.
I don't mind buying a cheapo USBasp like the one above, but buying another $20 clip seems a bit excessive for just this one project (especially too many what-if's with drivers from dx). How many of the pins are active for programming? Would it be possible to just use a few generic probe/hook type clips?