Brett, if you have a large USB (16GB) flash drive handy you can install the play disc on to that. I kind of need a new drive anyway, so I think I might snag one, use it as a dedicated GTA V drive until I finish with the game, and then put it into normal service. You can still install it on your Xbox hard disc if you really want but there will be a performance trade-off since you are sacrificing the bandwidth of your DVD drive.
Edit: I'm not one hundred percent sure what flash drive to buy at this point. It seems like any decent USB 3.0 model should do the trick as they are generally fast. The Xbox doesn't do 3.0 but they should max out the 2.0 interface, whereas that's less likely on a 2.0 device.