So I am heading to Alaska in July to do some hiking/exploring. Is it better to bring a firearm I own now, or buy a gun and ship it back here via dealers when I return? Looking at a S&W Alaskan Backpacker in 44 mag or 500. I have a girlfriend in Alaska that will shop for me.

And how many bears do you plan on shooting with that cannon?

Take something you are familiar with (don't buy-and-use on site). Understand it's idiosyncrasies. Find out the laws for there and where you live (as long as it's legal at each end then transporting within the US is protected). I normally don't like to fly but this would be a case where doing so could avoid shipping concerns (check with your airline for their requirements for checking firearms).