I responded to this by email back in June 2021, but I see that it did not log here. For future reference, the response was:
Unfortunately, the underlying ImageJ tools do not support fits header propagation, so the timing data gets lost when using "Grouped Z Project". However, the "Grouped Z Project" stacking capability with fits header support would be very useful for many applications, so we'll include this as a high priority AIJ development request (i.e. we'll add AIJ specific code to support this capability). It will require more that just coping forward one of the fits headers per stack. At a minimum, it will require calculating the center time of the averaged images, and probably other header modifications. There are tons of timing header values, so this could become complicated once we dig in. Nevertheless, we'll get something implemented, which is better than no timing information at all.
I can't promise a specific availability, but it is now on our list. In the mean time, I think your suggested Excel solution is probably the best workaround.