Of course there is an "other" observer - there is one at rest with respect to the clock, and one that is moving with respect to the clock (or sees the clock moving with respect to him)
. There is no time dilation within a single reference frame, it is always in one frame compared to another.
The person at rest sees one unit of time between the forward and backward transits of the photon, separated by a distance L. The person who sees the clock moving
does not see the forward motion take 2 units, because the distance is not 2L, it is 2L/gamma because of length contraction. Similarly, the backward motion is not .67L - you need to divide by gamma there as well. Once you do that (i.e. properly apply relativity) then the round-trip times are equal for any orientation of the clock.
The reason you must "tick-tock" is because of the asymmetry that you have introduced, and I explained above. You have to do an apples-to-apples comparison of the elapsed time.