Here's the deal, Sam5. The time dilation is not some all-on-its-own equation that Einstein threw in there. It comes out of those same transformation equations at the end of Section 3 - but it's not all that comes out of those transformation equations.
See, if, when viewed from the K frame, one clock starts out a coordinates (0,0,0,0) and the other starts out at coordinates (3,0,0,0) and they both end at (3,0,0,5), we conclude that the first clock moved (x-value changed from 0 to 3) at velocity 0.6c (it took 5 time units to move the 3 distance units).
However, that clock saw itself as stationary. So, if we apply the transformation equations to its starting and ending coordinates, we find that the starting point remains (0,0,0,0), but the ending point becomes (0,0,0,4). It stayed in the same place, but the time only took 4 units.
But we can't just apply those transformation equations to two of our relevant sets of coordinates. We have to apply them to the third set as well - the other clock's starting point. When we apply the equations to (3,0,0,0), we get (3.75,0,0,-2.25).
That's how we end up without a paradox. You want to simply apply the 80% time dilation without actually applying the transformation equations, and thus get a paradox. Because the 80% time dilation is a result of the transformation equations (but only one of the many results of the transformation equations), you can't do it that way.