Done it a few times, no big deal. Have a look at the chart (you can check it our on
https://skyvector.com/ by navigating to KGCN and then displaying it from the menu on top of the page). The tour traffic will be below you, that's the whole point - to keep the commercial sightseeing traffic separated from transients.
Depending on time of year, early morning is indeed better, as less turbulent. It can get quite 'interesting' out there
In winter usually not an issue. If you have (as you should) flight following, the controller will sometimes ask you which corridor you'd like to fly. Just be prepared for that Q. It's a pretty spectacular flight, but the only corridor that actually goes along the canyon is the Zuni corridor at the eastern edge of the park, the other ones cross it at roughly 80-90deg angles. Btw, while you are there, Monument Valley is nearby to the east and definitely worth a visit
