He said he couldn't agree more.
I think that "he couldn't agree more" in "he said he couldn't agree more" is a reported speech which requires verb shift back. Thus, the original sentence should read like this:
He said he couldn't have agreed more.
Am I right?
It may be easier to decide which to use, if you know the meaning, in time.
For example, if the sentence could be completed as, "Regarding the abolition of capital punishment, he said he could not have agreed more, until someone killed his father," then you will see there is a shift in belief. The "could not have" refers to something that changed.
If the sentence could be written as "Regading the aboition of capital punishment, he always said he could not agree more. Therefore, I was surprised to hear of his support for President Bush." His belief was firm, at all times in the past. It may have changed (there is a suggestion here that it did) but we've never heard him say anything else.