As the other posts mention, high density altitude operations can be tricky, and all too often, lethal for the inexperienced flatlander. If you must go, as Dream Land said, plan to go with minimum fuel, and the ONLY reasonable time to fly in and out of there is early morning. I live and work out of high elevation airports, and I would plan an arrival there before 10 or 11 a.m., and departure would absolutley be at first light, while it's still reasonably cool. Standard temperature at an elevation of 6600 feet is 2 or 3 C, nearly freezing, so any day between now and November is likely to have density altitudes, as you note, at well above 9 or 10,000. Further, it's essential that you understand proper leaning techniques, the usual procedure is to adjust mixture for best power (or a bit to the rich side) before take-off, as part of the run-up procedure. This is no different than adjusting mixture for cruising at 9 or 10K. The "Mixture-full rich" in checklists is written only for sea level up to maybe 3,000.
As you also note, takeoff and landing distances will be long, even for turbocharged aircrfat, there's a reason that runway is nearly 9,000 feet long, (and it's not because Singapore Airlines wants to send their A380 in there!) You must fly the aircraft by Indicated Airspeed, as always, but the view out the window on takeoff or final approach will be noticeably faster than you're used to, and somewhat disconcerting.
You do not mention where you will start this trip from, but, I would highly recommend a stop on the way, at a lower airport near the mountains, and hire a local experienced instructor to spend a day or more providing some high density-altitude training. A fully loaded Archer will have marginal performance, even for an experienced high-flyer. Every year out here where it's high and hot, there are tragic stories of overloaded aircraft crashing full of flatlanders on vacation. Please approach this wisely and cautiously so that you and your family do not join the sad list of high-altitude accidents.
You may also be better off to go with one of the professional scenic tour operators, rather than deal with the complexities of the Special regulations and airspace requirements over the Grand Canyon.
Hope this helps in some way, enjoy the trip, it's beautiful!