I'd leave the bear spray, chances of seeing a bear are slim, they are black bear, not griz, and even if you see one, it'll run from you faster than you can run from it, besides, you're going to be carrying a gun which is a better deterent than spray!
The most useful part of the GPS may end up being the ability to input the location of your downed elk. IF you elk dies in the middle of a forest at dusk, by the time you get it field dressed it will be dark, and when you walk out everything will look the same. Input the location of that elk so it's easy to go right back to it. Also, input camp location right off the bat. That may be the most important info. in your GPS.
The .257 is on the light end. Capable yes, pick your shots wisely!
Heavy bibs/parka usually aren't necassary. It's just usually not that cold that time of year. Lot of snow, possible, really cold temps., not as likely.
Your physical condition will be fine. Each person just has to pace themselves to thier own condition. Your strength will come in handy when it comes time to pack out the elk!