Not even visit the GC? One could stay in Tusayan, right outside the park if they really didn't want to stay inside the park. Taking an expensive helicopter ride over the GC (and you need to be sure if it's going over the main part of the canyon or the western part, outside of the park) is not the same as standing at the rim. Whether or not it is a madhouse, with nil parking, at the canyon in the park depends on when you are there and exactly where you go. It was awfully crowded the New Years Eve we were there, and snow and ice limited parking spaces (won't be a problem for you), but we're still glad we stayed there. If you can go during the week, and not on a holiday, odds of a peaceful visit are more in your favor. What is sometimes called the West Rim is ouside the national park and is the site of the Skywalk (which has bad reviews on Tripadvisor, but which I have not personally visited).

The North Rim, which is closed by snow from about mid-November until sometime in May, is far less crowded (and is higher and cooler and greener) than the South Rim, but would be a significantly longer drive in your case and I would not recommend it (now, if you could drive directly from LV and then on to the North Rim, it might be doable). North Rim has far fewer options for accommodations, however (the lodge at the rim, Kaibab Lodge just outside the park, and Jacob Lake Inn 40 miles fromthe rim; otherwise would need to stay all the way out at Fredonia, Arizona or Kanab, Utah). Again, if you give the dates of your travel, that would affect recommendations.