No. If a hydrogen atom makes a transition "there", then it does so with a frequency determined by the clock "there", at the point where the transition happened

No. If a hydrogen atom makes a transition "there", then it does so with a frequency determined by the clock "there", at the point where the transition happened. If we detect the emission "here", we see both the red shift and the time dilation between "there" and "here", and we see them in exactly the same way - as a frequency shift.