Mt. Cook Village is in a stunning valley surrounded by glacial mountains. The drive from the main road to Mt. Cook is gorgeous. The Tasman Glacier and Lake, which are closeby, are a bonus. So you will enjoy your visit. The road to Mt. Cook is not that high (it goes along Lake Pukaki) or particularly winding, so it does not close that often in winter, but it does close sometimes.
If you plan on staying in self-catering accommodation in Mt. Cook, you can pick up some fresh (farmed) salmon near the turnoff for Mt. Cook Village at Mt. Cook Alpine Salmon at the scenic lookout for Lake Pukaki. If you go to their website, you can see what the view looks like from Lake Pukaki's lookout on a clear day:
There are also more salmon farms south of Lake Pukaki, as you drive away from Mt. Cook. You might want to stop and have a look at the fish pens at High Country Salmon, which is right on the main road near Twizel.
If you add a day you can work Dunedin into your schedule, but it will make for a rushed trip. You would drive from Mt. Cook to Omarama then head to the Waitaki Valley/Kurow, then to Oamaru, then south along the coast to Dunedin. I assume you would like to go to Dunedin so that your kids can see the penguins. It will be 4 hours to drive from Mt. Cook to Dunedin, another 45 minutes to get out to the end of the peninsula (it's not far, but the road is challenging) . From Dunedin, the drive to Queenstown would about 4 hours in winter. Be aware that the roads can be icy in the early mornings. Here are winter driving tips:
Don't feel like you need to rush. If someone behind you wants to go faster, pull over and let them pass. There are frequent pull-outs and rest stops on NZ roads. By the way, if you drive from Mt. Cook to Dunedin to Queenstown, you can avoid Lindis Pass, a beautiful road but the second highest road on the South Island, which means it can be difficult to drive in icy winter conditions. The road even has signs stating that it has a "HIgh Crash Rate".
Alternatively, you can see the Little Blue Penguins at the colony in Oamaru, and which also has a few Yellow-Eyed Penguins at their Bushy Beach (though its viewing platform is currently closed due to recent heavy rains). You'd pass through Oamaru on your way from Mt. Cook to Dunedin. It is about 90 minutes north of Dunedin on the coast, It's a small attractive city with a old Victorian center that has been lovingly restored. You'd have another opportunity to see YEPs at Moeraki Lighthouse (Katiki Pt.), which is about 30 minutes closer to Dunedin (or 30 minutes south of Oamaru). Moeraki is also famed for its round beach boulders (a true geological oddity).
Before changing your plans, I would email the Blue Penguin Colony in Oamaru and find out what your chances are of seeing penguins in August (numbers are higher in summer). Remember, the Little Blue Penguins come onto land as the last light is on the horizon. The Yellow Eyed Penguins come in hours earlier, during the late afternoon. In winter, this would be from 3 p.m. on.
The Little Blue Penguins come onto land in large groups, the YEPs are largely solitary, though like the other penguins, they meet up with their mates onland. Here's a link to sunrise/sunset times in NZ.
I would also email Penguin Place on Dunedin's Otago Peninsula to find our what you chances are for seeing YEPs in August.
YEPs appear on most of Dunedin's Otago Peninsula beaches during the late afternoon. Sandfly Bay (no sandflies here) has a penguin hide.
Road condtions can quickly change in winter. Roads can close suddenly and you might have to take long detours. However, we can also have beautiful clear sunny days in winter. There's no way of knowing so far in advance. For the last five days, it has been gorgeous in Dunedin: sunny blue skies. It has even been warm at times. But two to three weeks ago, we had record breaking rains. Fortunately, I was in California. I returned to Dunedin to see the hills covered in snow (it gets cold here, but rarely snows). My husband remarked that the fields looked like flooded rice paddies (because it had rained so much). You just never know what will happen with the weather in winter (or spring or summer).
If you miss Dunedin and spend your last days just in Queenstown, I doubt you will be disappointed. It is lovely and there is loads to do, You'll be able to relax more, see more, and not have to drive so much.
You will probably need to book a few days ahead for Mt. Cook. I usually monitor accommodation availability on Wotif.com. It lets you know what accommodations are already booked up. When I see that things are starting to fill up, I'll book but I always check the cancellation policy first.