I did this a couple times in high school when I was just over, and it seemed to work (wrestled between 130 and 145). The key was always to get on the scale right away. I've also see guys weigh in holding their hands in the air right as they step on the scale.
I'm not completely sure why it works, though. I could see your coach being somewhat right about the blood flowing downward. Your weight is based on the acceleration that gravity places on the mass of your body. When you stand on your head, there is a higher concentration of mass toward your head than normal. When you go right side up, that blood takes a little bit of time to flow downward because of gravity.
Although the blood flowing downward would not be nearly as fast, look at it this way. Imagine you are standing on a big scale with a bucket of water and a really tall tower. If you hold the water in the bucket, you weight Wtotal. The water weighs Wwater. Now, if while on the scale, you dump the water out of the bucket while up high and into another bucket on the ground. Before and after the pour, the weight the scale reads is Wtotal because the water is included in the system. However, while the water is in the air, the weight will be Wwater less than Wtotal because gravity is causing movement of the water, and there is no reactionary upward force.
Not that your own blood leaves your system when weighing in. But the extra blood in your head will actually flow downward instead of staying relatively constant like your body normally does.
Does this make sense? Or am I overanalyzing it? I would probably never have believed it if I hadn't done it myself.