That's a really good question. And I really can't answer it because since I don't believe the earth is accelerating at close to light speed.
You're the one who made the assertion. It certainly should be your job to make it complete, or to rescind it.
I would suggest that the frame of reference you've implied (yet can't identify) does not exist outside of a hypothetical thought experiment. I'd be curious to see if you, the claimant, actually put any thought into your claim, or if you just rapid-fired it with its glaring holes.
Perhaps you or someone else who does believe that can answer the question...relative to what is the earth accelerating?
An observer immediately above the Earth, at rest relative to the Earth.
My point is that if the earth is accelerating at c (or really at any rate at all), any satellites would have to be accelerating at the same rate to keep pace. Any faster or slower, eventually it would be out of functional range.
This, too, is incomplete. In order to fulfil your requirement of the satellite not escaping or crashing into the Earth, it has to be accelerating upward together with UA. However, this does not mean that it can't be moving (or accelerating) perpendicular to UA for periods of time, or even oscillating up and down irrespective of UA. As long as this motion remains cyclical, your conditions can easily be met.
If the earth and GPS satellites are accelerating at the same rate, there wouldn't be any time dilation.
And this, too, is incomplete. The multiple relativistic effects experienced by GPS are (primarily) due to relative velocity and a difference in gravitational potential. Your argument might hold some water if the satellites were geostationary, but they're not.