It depends, in part, on how large the bear is you shoot. Since they come mega-sized, your plans should include the possibility of a monster. When I hunt brown bear, I prefer a .338 or larger caliber bullet. If I had to hunt with a guide, I would want to shoot and kill the bear with my own shot(s), rather than having the guide kill my bear for me. If your bear isn't anchored pronto, your guide is most likely going to cut loose until the bear is no longer making headway. So if I were hunting with a guide, I'd shoot no less than a .338 with an effective bullet guaranteed to expand, and guaranteed to retain the majority of its weight.
You'll have no way of knowing how far the shot will be, but your guide will most likely want you within 200yds. Discuss it with your guide, but most prefer closer than farther, such that they have the best chance of putting the bear down with follow-up shots, as necessary.
Ask your guide what he carries and what most guides he knows carry. Not many use a .308 caliber. .338 is generally the minimum. 375 may be most common. Some 416s, and even 458s.
If I were going on a guided hunt, I'd take at least a .375. The rifle will cost a lot less than a guided brown bear hunt costs these days. You might as well improve the chances you'll kill your bear, rather than coming back knowing your guide killed your bear. I know of one guy that didn't stop his even with a .416 magnum. His bear was spooked before he shot, which makes a big difference in my opinion. He drilled it thru the chest twice before the bear got to the alders. They waited like six hours before going into the alders. The boar was still alive and came after him. He did finish it off. It was a 10 1/2 to 11' boar, from down on the Alaska Peninsula.
If you want to be really certain he'll go down and stay down, use a .458 Win Mag with a good expanding bullet at less than 100yds.
They can be scary big the first time you see a large one at less than 75yds. Here's a photo of a skinned 10 1/2 foot boar I shot on Kodiak Island. Notice the size of his skinned front paw relative to my hand. I'm holding only one of his toe pads.
)
