We'll you had to bring bear into the fold eh....
My ultimate choices would be the 8mm Rem mag or .311 Imperial Mag for the deer through bear thing...
Flat shooting through long ranges..Perfect for goat and sheep...Yet pack more than enough punch to dump a bear of any size......
I personally use a 30-06 for all my hunting deer, elk ,moose sheep, goat...
And I do carry a load of 180gr..TBBC for bear just in case I find myself in trouble......
I do have a .338 but they are a real pain its a real pain in therear to carry around on long hikes and trips deep into moutain terrain...
Therfore givin another option I'd rather carry the wieght of either of the 2 cals. I mentioned above...
The 225gr. TBBC in a 338 really isn't flatter shooting than a 180gr. in an 30-06...Velocities are roughly only 100fps 125fps apart depending on propelents used...
Anything less than a .225gr. in a .338 and you might as well shoot a
.300 win in 180gr.......
Therefore my answer to your question...,300 win mag with 180 gr..bullets is my choice for deer through bear...