<If you like the lens try it in DX mode on your D800, its still an 18 mp DX camera. I even use my 70-200 2.8 VRII in Dx mode occasionally to get a little extra reach on my D800E.>
In DX mode your D800 is much closer to a D7000 in terms of ISO performance and file size. You really are giving up quite a bit.
As Brian suggests, crop mode is just cropping the image to a smaller size. There is no advantage to using a 70-200 in crop mode vs. cropping after the fact (other than file size). What's more, you are using a full sized viewfinder so it is not as easy to see as a DX viewfinder.
Sure you can use the 17-55 in DX mode and it will be roughly equivalent to the 24-70 in some respects. But there is a big difference in your images. The relative depth of field is about 1 1/2 stops shallower with an FX less on an FX camera - check the DOF tables for specific examples. The FX image size gets the benefits of large file that is often downsized for printing - and you lose much of that advantage in crop mode. If budget is your issue, the 24-85 is readily available and will better fit the capability of the camera than a DX crop using a 17-55.
Quality wise, the 24-70 is modestly better than the 17-55. But the other differences are very important. This is not a close call or matter of opinion as with many lens choices. The 24-70 clearly is able to take advantage of the D800 in a manner that is not possible with a 17-55 in crop mode.