Not meaning to answer for @moondog, but usually it's a bad idea for logistics reasons due to the geography, particularly when our usual GW recommendations are for Mutianyu, Jinshanling, or maybe Huanghua or Jiankou. The route you would take to get between any of those and the TofH puts you squarely in Beijing's worst traffic jams, and travel time could easily take an extra hour to get through that. Hence my suggestion of Badaling GW which I think is reasonable in certain situations and at certain times of the year. Your visit happens to fit this. While it would by no means be traffic-free, it is more direct and this part of the GW is closer to the city. And coming back into Beijing to the TOH, the middle of the day is not quite as bad as morning/evening rush hour, hence my suggestion to put this in the early afternoon.

If your co-worker has an itch to see the GW, I suggest you go ahead and scratch it. Certainly I'd rank it higher than the Summer Palace at this time of year, even if it is Badaling and even with the logistics. If you cannot come to an agreement, then maybe she goes to the GW with the driver (doesn't really need a guide) and you can go to the Summer Palace with or without the guide, and on their way back in, they can pick you up at an S.P. exit to go to lunch and the TofH.

While I would get a relatively early start, like leaving Beijing around 07:00-ish, in winter I don't see a lot of point to get up before dawn and trying to get to Badaling much earlier than 08:00. Certainly I'd try to get breakfast at the hotel before you leave, assuming it starts serving at 06:00 and is included in the room rate.