Bradley,
In that case just go for a walk-in.
If you haven't been to the GC do the classics – Bright Angle and Kaibab.
Assuming you have the legs for it…
Down the Kaibab, which will get you in ahead of the crowd and a good choice of campsites, to stay at Bright Angle campground for 2 nights. If you want to experience the Canteen family style dinner ($25 each for Steak or Stew) sign up as soon as you get there. (But do take a dinner for that night in case they are full.) Spend your second day hiking up to Ribbon Falls. After dinner walk the river loop. Return up the Bright Angle enjoying the Indian Garden area as you go.
Or some variation.
I have done walk-ins on numerous occasions with my wife and with other friends. If things are busy, which is likely in March, plan on spending 2 days being a "Rim Tourist", or spend a day walking halfway down the Hermit and back, or Grandview/Horseshoe Mesa.
As soon as you get to the GC go to the BCO (Back Country Office) and ask for a permit for the next day. If they say "Sorry", talk about options. Then return the following morning.
Go Early to the BCO before it opens, stand in line and you may get your wish. Ask about options. If nothing works, you will get a number for the following day. Do not be discouraged by a high number. You'll be on your way by 10am. Go be a tourist.
Go back to the BCO the following day and wait for your number to be called, and plead your case. If you don't like what you hear, ask the ranger for suggestions. There is a 99% chance you'll get your 2 nights at the bottom. Go be a tourist again. Then pack and go. There is a bus that will take you to the Kaibab trailhead.
It's not wilderness trip but it is a great place with a wonderful history.