A proper treatment should use GR but here is my reasoning; as I said before it is importent to understand what is meant by measuring times and lengths - you should really measure events and intervals as these would be invariant. Please bear with this lengthy description.

I prefer to use two observers - one at a long distance from the earth and another at the centre of the earth (ignoring practical difficulties that do not change the gedanken experiment). The reason is that neither will be in a gravitational field (or at least not one that cannot be made arbitrarily small) but the one at the earth's centre is at a lower gravitational potential.

The photon clock is constructed of two miirors spaced apart by a distance "x" so that a "tick" would be 2x/c long. In taking such a clock to the centre of the earth you, as a local observer, would not see any change in its behaviour. The mirrors will be the same distance apart however measured with any method you would have locally and the ticks would appear at the same time interval as when you constructed it (using measuring sticks, other clocks etc).

To a distant observer though, you and your clock have moved to a lower gravitational potential. The ticks from your clock (and any other clock) will be redshifted and running slow compared with his clocks. The apparent conundrum is why, if the velocity of light is the same to all observers, should the ticks from the photon clock be slow? The answer is (I think) that, at the lower potential, lengths (as measured by the distant observer) are larger so the mirrors are further apart. It is importent to understand how a distant observer would measure lengths and this is fundamental in solving even some apparent paradoxes in Special Relativity. The distant observer could measure the mirror spacing by sending a light pulse that partially reflects of the back of the first mirror (closest to him) but also passes by and reflects of the front face of the second mirror (furthest away from him). He would receive, some time later, two pulses with a time interval between them. This time interval (at the centre of the earth) would be the same as the tick time, but it would also be redshifted in returning to the distant observer. Like the tick, he would see that the gravitational potential gradient would have stretched the measured time to 2x'/c. This is interpreted exactly in the way that Lorentz contraction in SR is defined. It is the only way a distant observer can measure distances and lengths. The mirror spacing is larger to the distant observer.

The correct and more rigorous treatment would use GR and intervals in space-time but is mathematically challenging - at least for me.