At one point I had read that one could travel thru Illinois and carry in your vehicle under the laws of your resident state. But, you CAN NOT leave the vehicle while carrying. I would do some further research on this, I have seen this in a posting from the Illinois State Police in the past. Travel in and around Chicago with a firearm at your own risk as they operate under their own rules there! Cities in Illinois can make their own rules, no state preemption. Illinois Highway Patrol has NO SENSE OF HUMOR, and I personally believe they actively look for out of state plates. I don't believe I would inform them I was carrying, nor would I exit my vehicle unless I was drug out.
As a Wisconsin resident who travels to Chicago weekly, might I make a suggestion. Depending on where you are going in Michigan, consider taking Hy 29 across Wisconsin to Green Bay and north to the Mac Bridge and avoid the Chicago/NW Indiana Zoo entirely! Much better scenery, and maybe a few hours longer IF you don't get caught in traffic. The farther north into Michigan you are going, the less out of route miles it will be. (Fringe benefit, you're a mile from me and I'll buy you supper if I'm home. Abbotsford)
If you don't want to do that consider staying as far out as you can. There is no real way to avoid I-80 across if you are going to Michigan, but look at coming in on I-90 to 290 to 355 south, 88 east to 294 south to 80 east and that keeps you out of the worst of it. Also consider getting an I-Pass or E-ZPass as that will save you from stopping at all the toll booths. Go thru Rockford instead of Milwaukee. Construction on I-90 in Illinois is done west of 290, Milwaukee is a mess. The 290 east to 294 east interchange near Berkeley is a disaster during any daylight hours, avoid at all cost. Too many things happening, too many trucks slowing things down. The whole area is really a crapshoot, traffic jams can happen anywhere, any time, for any reason.
Another option would be to take I-355 all the way south to I-80, about 15-20 minutes longer.
Or, take I-39 out of Rockford south to I-80 about 45 minutes longer.
Keep in mind it is very easy to waste 30-45 minutes in a traffic jam around Chicago, with no real way to see it coming. Pick your poison basically.
If I haven't been clear, I HATE Chicago! I spend absolutely NO money there except for tolls.
Fuel and food in Wisconsin before you leave. I've also found the restrooms along the Illinois toll system to be notoriously filthy.
Edit: There is also a ferry that travels from Manitowoc (south of Green Bay) to Ludington, MI. Not sure of winter schedule, and I think you need reservations, so maybe check that out, but don't just drive there and think you will get on.
https://www.ssbadger.com/