Certainly could wait the before/after hours route but that kind of defeats the whole "meeting the family up top for a picnic lunch" idea. Again, you could "renegade it" and hope that you don't get caught if you're into that sorta thing (I'd recommend against it personally), but the I'd stay off the road while traffic is on it. I've actually driven on it a few times to during the tourist season, and people are ALL OVER THE PLACE on that road between them looking at the views and the "general" touristy, mountain road, nervous/bad driving factor, it's a mess.
If you think a twisted ankle would mess up your hike, try getting struck by a passenger. Not kidding here at all (no smiley at all). I wouldn't be caught anywhere along that road while "tourists" are driving up and down on it.
If you really MUST do the "road route", prolly the easiest way is to register for the
WHITEFACE MT. UPHILL FOOTRACE and just run UP and DOWN the sucker. For 35 bucks, you get periodic water stations and a BBQ dinner up top too (families can meet you). Plus, looking at the results from last year, AT LEAST 30 of the participants were at 15 Min/hr (4 MPH), which on an 8% paved grade, is a reasonable "brisk" walk. I'm sure that some participants walk/run down too since not everyone would have a "ride".
Would it count for the 46ers? Don't know. Can't see why not. Would is count for you personally? Sure, why not? It would for me.