The 500 lbs of gear will most likely be taken out of the Tahoe and put in the trailer so figure passengers and fuel for the payload and that will let you know about max tongue weight.
I pull a 7k # trailer with an older Tahoe out of the Denver area and almost all in the mountains of CO. 2 to 3k miles per year. Your Tahoe will pull the trailer you are describing just fine IF you do your part. The bigger the trailer the more you are going to have to work while driving. Not just put it in D and go.
As has been said you need to keep the RPMs up while climbing but it's not a bid deal and soon your family won't notice 3500 to 4000 rpms. Use the manual gear shift to hold it in the gear/ rpm range that will work for you and don't let it shift every time you level out for a few seconds or let off the gas a little. The hills aren't that long and if you slow down it's not that big a deal in the grand scheme of things. Also you'll notice it shifts very hard going up hill with a load so it's better for the transmission to hold it in one gear at an RPM which will get you all the way up the hill. You'll notice that when you hold it in gear and use the rpm's you won't even have to give it too much gas going up the hills.
Planning ahead for stopping, accelerating, passing and even changing lanes becomes more important and fatigue can set in if you aren't used to it and is worse with a heavier trailer so I would worry less about the Tahoe's limitations and ask yourself what you or the other drivers are comfortable with.