There are too many holes in the thought experiment. Firstly, you can't have a photon oscillating - photons don't have classical trajectories, as has already been stated in this thread. Secondly, to measure a photon's frequency, you destroy the photon. Third, you have to specify how the data signal is encoded, otherwise we couldn't tell you whether relativity will have a relevant effect on the data stream. Fourth, *every* photon emitted from the moving object will be Doppler shifted, not just the one you're talking about, which as I have already said, was destroyed when you first measured it's frequency. The thought experiment you have proposed doesn't make any sense.
Next, you make it sound like you think time is some fixed thing that is the same for all observers. It isn't, and it can't be. If a clock moved at the speed of light (thought experiment only) relative to some observer, the observer would never see the clock tick. Ever. Similarly, we can never, ever see time pass for a photon. To do this, we'd have to be in it's frame, which is relativistically impossible, since *every* inertial observer must measure light to travel at c in a vacuum.
On the other hand, we definitely can see time dilation for moving objects with v < c. So, for example, if you had a ball on a spring oscillating, and some device measuring it's oscillation frequency, moving in a spaceship at a constant speed relative to you, then you would see a reduced frequency. However, this has nothing to do with how the information is transmitted, it is a property of the Lorentz transformation between the frames.
Finally, I would like to say that I suspect the point Feynman was making was that with photons *it does not matter* what a photon 'does' in time - there is *no* frame in the universe from which we can observe a photon 'moving in time' - the instant we know a particular photon exists (say, when it strikes a detector), it's gone. It's the ultimate shock therapy.
The problem with this discussion is that quantum electrodynamics deals with photons on a different footing from the 'wave packets' of classical electromagnetism, and usually when you're discussing relativity on it's own you talk about 'waves' and 'particles' having classical properties. The unification of SR, EM and QM brings new insight into the problem, but it goes way beyond the scope of the colloquial physics here.
Cheerio,
Kane O'Donnell