There is a campground 4 km from the lake, Lake Louise trailer campground but only has electric, no water. We have always stayed at Tunnel mountain trailer court, which really isn't what I would think of as a "trailer court". It has full hook ups , lots of trees, all pull thru. Close to Banff and very nice.
As far as crossing the border, no problem with passport. Your rig may be X-rayed coming back into the states. If you have pets they may ask for shot records going into Canada. Also keep pet food in its original packaging. Coming back one time, border guard seen our dog, asked if our dog food was in its original package, which it was not, told us that was a no no .