Your question is quite well-formed, however all I can say so far is: I don't know. Maybe it is known, however I could not find any information on this in the literature. In general, little is known about the crosstalk between circadian rhythms and apoptosis. It is known that circadian rhythms regulate DNA damage repair, so I would assume that the same is true for other maintenance functions such as apoptosis of redundant or damaged cells.
I don't know which research tools are available to you, but if you want to look at the effect of the clock on apoptosis signalling in a simplified setting, I would recommend synchronising the clock of culture cells with dexamethasone (you'll find a lot on that in the literature) and then using the apoptosis read-out of your choice on samples of these cells taken in regular intervals. I recommend taking samples every 4 h over the course of 24-48 h. You can make this less tedious by synchronising separate batches at different time points. As read-out for apoptosis I recommend Caspase-3 / -7 activity, ideally by Western Blot or an enzymatic activity assay using Ac-DEVD-AMC or something similar.
If you want to have a look at the effect in a more complex, physiological model, you will have to use mice.
I hope I could help you a little. Good luck!