Your cross pull would be 19' 9-5/8". Unless the ground is very uneven, it shouldn't be too hard. To use a transit, you could setup in the middle of your square, set zero due "north"(assumed), then turn 45° increments, and pull 1/2 the above measurement, but that complicates things, and doesn't make measuring much easier.
Alternately, you could setup on one corner, sight a corner on line, and use 0°, 45°, and 90°, and pulling the appropriate distances to setup the square. That would be a good check after using tapes only. Setup the square using tapes, then setup the transit and wind angles. You don't really have to measure then. Your angles should hit the points you taped to.