You're the captain/pilot, you decide! There is no particular right or wrong. You are advising ATC of the priority required, as
you assess it.
I tell 'em the guy on the ground is of no immediate use, so fly the aircraft first, do the memory items next and then decide what they need to tell ATC and the cabin crew.
Whilst I agree that there is a priority of requirements, ATC might be glad to know quickly that you possibly shed bits of your motor all over the runway before he/she clears the aircraft that was waiting behind you at the holding point to take off, or the inbound to land, into a potential debris field.
An early call of
Mayday (or
Pan if you really feel that is more appropriate) should fit easily into the time available after the problem has been initially identified.