The Rolex Cosmograph Daytona is a certified chronometer chronograph as is the Omega Speedmaster Moonwatch for example. They are not unusual. In theory yes, anything connected to the time train could affect the timekeeping especially if the use of the chronograph function is intermittent and unpredictable (as it is). This is why regulator accuracy clocks do not chime or strike for instance (the time train tripping the strike train could affect accuracy). I do not have the reference but in pocket watches I have heard of watches with two completely independent time trains for the time keeping and chronograph (I have never actually seen one of these but have read about them). These watches would have 2 mainsprings, 2 balances etc. I do not know how the chronograph function is tested in relation to the current certification process for chronometer accuracy.
Flyback and split-seconds chronographs aside, the two main types of chronographs being the cam and column wheel systems. Every time you start the chronograph you connect it to the main caliber to pull the necessary energy, so yes it can have a small influence on the movements precision. COSC certification (and others) as far as I know don't factor in complications and other functions of the watch, only accuracy, but I could be wrong. I do know that many manufacturers have abandoned these standards and opt for in-house tests and trails. Interesting question nonetheless.
Given that automatic chronographs didn't exist until 1969, and there were very few manual wind chronograph movements certified as chronometers prior to that time, this is the earliest examples of an automatic certified chronometer:
This was made for the 125th anniversary of Louis Brandt et Frère (later known as Omega) in 1973. To my knowledge, the standard manual-wind Omega Speedmaster Moonwatch was not certified as a chronometer.
Many thanks to all who helped with this inquiry! A number of people have told me that the ordinary chronograph wristwatch is used so infrequently in the chronograph mode that the extra power draw would likely have almost no effect on the chronometer performance.