What a wonderful question!
You've put your finger on something very important. Essentially, because the transmission time in this case doesn't change, the two observers can't agree to disagree about whose clock is faster. Regardless of what one might say about who is "right", as you argue they must agree about who is fast.
The key is this. The Special Relativistic arguments only apply to a non-accelerated observer in a region where gravity is unimportant. Just to go step-by-step, let's say there was little gravity here and the satellite was kept in orbit by a string. The satellite is accelerating with respect to the earth. Therefore SR by itself doesn't describe what things look like in the satellite frame. However, we can then deduce, using your argument, that its acceleration towards the earth causes earth clocks to appear sped-up, by twice the amount that the SR effects cause them to appear slowed down (all this is just to lowest order in the effect). Then the apparent speed-up of the earth seen by the satellite will match the apparent slow-down of the satellite seen by the earth. This is the way, in some of my course notes, we derive the (lowest order) magnitude of the effects of accelerating frames, the start on the path to General Relativity.
Now when you consider gravity as well, another ingredient (the Equivalence Principle) is needed. It turns out to make the object higher-up in the gravitational field go slower. For low-orbit satellites, the first effect dominates and their clocks are slow. For high-orbit satellites (e.g. geosynchronous) the second (gravity) effect dominates and their clocks are fast. That's important for the very precise clocks used in GPS systems.
(published on 09/29/2010)