Hi lunren,

Typically comparators have a predriver stage, which is NOT the one introducing the largest delay in the circuit. You could use a pre-driver stage in parallel with the main one that switches a fast signal to trigger your counter, or eventually use the same pre-driver you use in your main com, for example looking at the pre-driver's stage output and using that signal to also trigger you counter.
Obviously how easy is to implement this will highly depend on the topology or your comparator.
Whatever you use will have to be much faster than the delay you want to measure.

Best
Tosei