However, the symptoms which have been perhaps been overlooked, is that the Op is feeling stressed or depressed. These are both Medical conditions and therefore if he is genuinely suffering either condition, or even another condition brought on by them, he is quite within his rights to claim sickness and/or get help for them.
I am very close to my pet, and they are very old, will I take a sickie when they die - I doubt it, but then we are all individuals and we are all affected in different ways. Still if people want to complain about RMs mechanistic approach to "their people", but then be guilty of doing the same thing in this case, I won't be so crass as to point it out.