@wnb5th
My recommendation is to aim for either mid to end of November, or mid to end of December, in SD, if you can. It depends on when the colder weather hits, but early December can often be a transitional period where it is a bit harder to find the birds. Not that you can't find them, but I've personally seen more birds before and after the transitional period than during it.
A skim of ice allows the birds to hang out in the cattails, but not us 2 legged predators. It can also be a bit riskier for the dogs.
Mid to late November will generally give you warmer weather. With kids, and coming from a warmer climate, that could be a very important variable.
When the cold does hit, and we get a few inches of ice in the sloughs, then the birds will be more concentrated, and accessible. But again, it might make for a miserable time without being acclimated to it. Smaller bodies (kids) may not keep their core temperature up in some conditions as well as larger bodies (adults) can. And even though I love hunting small pothole sloughs in winter, there is some risk of going through the ice. I'd be remiss in not mentioning that.
No guarantees, but once the ice is generally safe to walk on, I find it safer on the non-cattail water, than amongst the cattails. Of course, the water is often deeper outside the cattails too, so there is that. It depends a bit on how thick the cattails are and what is going on under the surface. I am talking about bodies of water with fringes of cattails around the edges. I personally feel better, that time of year, with sparsely populated cattails patches, where you can walk around the clumps. It is harder for the birds to evade in those conditions too.
There will be plenty of birds in some of the big sloughs, if there is a food source nearby. Even if the ice is several inches thick, and relatively safe, some of those sloughs can still be miserable walking, even for adults. And if the cattails are tall, it would be counter-productive. Smaller potholes are much more manageable.
An option for pay hunting might be a western MN game farm, where you pay per bird, vs pay per person/gun on a SD game preserve. You don't need an out of state license. You get a receipt to prove the birds harvested aren't wild.
Some places will want you to pay for a certain number of birds per person (3 is common), but if you explain you have kids along, they may relax that a bit. An option would be to try your darnedest in SD, and if things don't go as well as you'd like, you could hit a MN game farm on the way back. You'd generally want to call them a day or so ahead of time to get a field reserved. You can also tell them you'd like an easier walking field for the kids, etc. Some are rooster only, and some have both sexes available. The other reason I mention it is your desire for a mixed bag. Some might have chukar and quail available, in addition to pheasants.
One last bit of safety advice, for if you do go for some pen raised birds as part of your hunting... There is a greater chance of low flying birds with pen raised birds than wild birds. Obviously, wild birds can fly low too, but there is a still a greater chance of it with pen raised birds. Farm raised chukar are fairly often low flyers too. A dog bounding after a bird flying away from you, or a slow crossing flush, can present a safety hazard for the dog. Kids also may not have the same situational awareness when swinging a gun at a bird that flies to the side or behind the line of hunters. That issue is magnified when dealing with low flying birds. When hunting with children or people not used to fast dogs that jump high, I always discuss this issue and get a firm agreement not to shoot at low flying birds -- and definitely not to shoot at any birds on the ground. I am not trying to preach here, but simply pointing out some variables to consider, since I don't know the background of the people involved (and others will read these posts too).
This is my first year of not hunting with one or both of my 2 daughters, now that the youngest is a freshman in college and busy with other endeavors, since 2011, when my oldest started hunting. We would hunt opening weekend together every year. Enjoy these years and have a great time together on your adventure, regardless of how many birds you see or bag.