I would go with the 10mm Auto in a carbine before I went with a .45, esp if bear was on the menu. A 220 gr hardcast can be driven from a 10mm handgun at over 1100 fps, so from a 16 inch barrel you could probably add another 100 fps or more to that, and that would be putting you in roughly .44 Mag territory. Both the 200 gr and 220 gr 10mms have a higher sectional density than a 230 gr .45 caliber round, which means all other things being equal, they should provide better penetration. Being able to get a 200 gr JHP to over 1400 fps for hunting and defense, and a 220 gr hard cast well into the range of legit .41 Magnum makes the 10mm Auto in a carbine much more potent and useful than a .45 carbine. The .45 just doesn't have the testicular fortitude to compete.
The 9mm operates at enough pressure to give decent bang for the buck from a carbine, so if it was just fun or home defense, or if the price of ammo was a consideration, I might vote for the 9mm. But if you need more power than the 9mm from your carbine, I would vote for the 10mm Auto over the .45 Auto.