SHORT ANSWER
FumbleFingers's comment and oerkelens' answer prove by giving counter-examples that neither "could not have it" nor "could not have had it" implies an attempt.
LONGER NON-ANSWER
Here, what I'm trying to describe is in which way "could not have it" and "could not have had it" are different. I'm doing so because I find interesting the use of "could not have had it" when two references of time are not explicitly given in the context. Let's consider both options:
(1) He had a clear idea – he needed seven years of peace and quiet in order to work. In America he could not have it. So he went back to Russia.
(2) He had a clear idea – he needed seven years of peace and quiet in order to work. In America he could not have had it. So he went back to Russia.
I think the key to answer this question is not in the modal verb could, but in the use of the perfect aspect: have it vs have had it.
Whenever the perfect aspect is used in a sentence, it makes us think of two instants in time. For example, in:
I have been reading for two hours (present perfect)
the two instants in time are now and the moment I started reading.
However, in
He could not have had it (past perfect)
what are these two instants? In the quoted text, there is only one reference of time. When a second reference of time is missing, my mind defaults to the present. So when I read "he could not have had it", I understand that something could not be had in the past, but in the present it could be possible.