Being a fellow Michigan hunter down in the shotgun zone, I sympathize with you.
I all most went to a 375 win. I decided to play around with the 357 max but didn't get out long enough to develop a good load this year. The gun dealer that I frequent uses a 375 and raves about it for deer. That said, I use a 44 mag launching hornady 240 xtp's in the mid teens. It gives me a 13 inch drop at 150yards with a 75 yards zero.
Now that S&W came out with the 460, if it were me, I would take a good look at it. As I under stand it, zeroed at 125 yards it wont shoot above 4 inches at closer ranges and at 200 will drop 4 inches. The BIG plus is you can shoot 45 colt and 454 cartridges out of the same barrel, no modifications needed.
I have a 454 barrel I bought the year before the 460 came out. I occasionally use it for deer as well. But have always been content with using 45 colts rather than 454. It shoots great, about 1 to 1 1/2 at 50 yards for 3 shoots using 250 xtp's around 1300 fps. Both the 44mag and 45 barrels have put deer down without them traveling more than 10 yards from point of impact.
Good luck on your decision. You will do well with either the 375 or 444. The 375 may shoot a little flatter than the 444, but they will both work great.
You can plug in the info on hornady's web site to find out which would shoot flatter, under the external ballistics link. You will need to know approx velocity and B.C. of the bullet and the program will give you a bullet drop chart to what ever range you want. Of course all ways verify these, but I have found it very helpful and very close to actual performance. You can find the B.C. of all Hornady's bullets under the bullet look up and the info section of each bullet.
If you order a custom barrel, I would have them put the brake on as well, may say you money in the long run. Just remember these will greatly increase the concussion to you, the shooter. Ear protection will become a MUST for any shooting...