"Stopping power" is a bit of a myth. The ONLY way to truly "stop" a big game animal is with a shot to the brain or spine. And just about any cartridge from .223 and up is capable of that. What you want is a cartridge/bullet combo that will effectively penetrate heavy hide, muscle, and bone and do damage internally to the vital areas that are a much larger target area than a head or neck shot. The only advantage magnum cartridges have over non mag is range. They can push a bigger bullet faster. Take the 300 win mag and the .30-06 for instance. Out to 400 yards, the .30-06 is going to kill a Moose just as easily and just as dead. Past 400 is where the more powerful .300wm is going to shine in difference. And I can tell you from many years experience, Moose aren't in country where you will be presented with many, if any, shots over 400 yards. If you or your guide is any good at all, you wont need to shoot past 100 yards to be honest. Bull Moose are pretty much the dumbest of all the deer species and most arrogant. Their sheer size makes them far less wary than anything else. So they are actually quite easy to get in on.

Now, if you are talking Brown Bear, that's a whole other hunt. You can be presented with shots out to 500 or crap your pants close. If intentionally hunting them, I'd opt for the .300wm or even go for the .338wm. I've been up close and personal with Brown Bears charging me and dropped them with handguns. But I was one LUCKY fella as I have trained consistently and often with handguns all of my life and am a VERY good shot under pressure. My recommendations for Brown Bear hunting is shoot them with the biggest cartridge/bullet combo you can handle at the furthest range you are consistent with

As far as rifle brands, that is a personal choice. Your best bet would be to handle several different makes and models and find one that fits you well. Factory rifles in your price range are all about the same performance wise really. In your price range, my nod would go to one of the Savage models, this one actually in .30-06, http://www.savagearms.com/firearms/model/11FCNS , and top it off with some good glass such as Nikon, Leupold, Vortex and you will probably be coming in under your $1500 range.