The writer wants to express:
We wanted to carry on, but it was not possible.
If they had simply written If we had carried on ... it would have expressed only that they did not do so.
In this context, could + not + have + [past participle] is used to say that something was not possible, or that it didn't happen (was not "real", actualized, or manifested). The not is implied in the conditional formation.
"There is no past tense, but could have followed by a past participle is used for referring to something in the past that was not real, or something that may possibly have been real:
I could have been killed.
What was that noise? Could it have been the wind?"