The barnes bullets can easily handle 3k fps. They are extremely tough bullets and are a whole different world than the cup/core traditional bullets that originally had trouble in the 378 weatherby. Penetration is very rarely a problem with the TSX or similar mono metal bullets because of their excellent weight retention and sectional density. The last moose I shot with my 300 RUM and 168gr TSX I got complete lengthwise penetration, 5+ feet in a bull moose, and moose are much heavier boned and muscled than a brown bear.
Bear, even the biggest brown bear, are not really that heavily boned. They aren't even close to the bone size or density of the big african game like zebra, cape buff, hippo, and elephant. A solid isn't the best choice for bears. An expanding bullet imparts a tremendous amount of shock and secondary wound damage that you simply don't get with a solid. The internal damage from a fast 250-270gr TSX will be more than any of the cartridges you mentioned.
I'd stack up the faster 375s with appropriate bullet selection against any of the proven african stopping rifles, 458 Lott, 416 rigby, 404 jeffery, and think you would be hard pressed to find any of them any more effective. The 375 RUM will also push a 350gr bullet to 2600+ fps easily. and a 350gr .375 has a much better Sectional density, ballistic coefficient, and penetration potential than any 350gr .416 or .458 bullet.