Honestly, it sounds like you're tempting fate here. Taking just a bivy might work...but then again, it might possibly ruin your trip or (worst case) put you in jeopardy.
If you are just getting into hiking and packing light, don't make the mistake of thinking you need to have 'ultralight everything' until you've had time to test and experiment with your gear in more favorable conditions. Pitching a tarp properly will take some trial and error, especially if conditions are less-than-favorable. I wouldn't want to try to learn those skills on the fly in a place like the Grand Canyon, where it can be a challenge for even experienced hikers.
My advice would be to get a cheap one man tent that you can count on to be watertight. It will be heavier than a tarp and bivy, but you've got all the time in the world to try those systems out when you can do it right.
If you're sold on the tarp/bivy idea, then you can always make your own tarp practically cost-free with some plastic sheeting or tyvek. Use a groundcloth, read up on site selection, and practice pitching it without driving stakes into the ground until you can get a good pitch quickly, hopefully in windy conditions.
Don't go ill-equipped just for the sake of going 'ultralight' and by all means, put yourself in a postion to enjoy your trip as much as possible. The Grand Canyon is one of the most magnificent places to hike on the entire planet!