The CCA can be defended in more than one way. With some modest Leibnizian principles, it's fairly easy to show why the fallacy of composition charge against the CCA simply does not hold up.
1. Everything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause. (Premise, PSR)
2. A series of contingent causes exists. (Premise)
3. Hence, the series of contingent causes has an explanation of its existence. (From 1 and 2)
4. The series of contingent causes does not exist by a necessity of its own nature. (Premise)
5. Hence, the series of contingent causes has an external case. (From 1, 2 and 4)
6. Therefore, the series of contingent causes is explained by a necessary cause. (From 1, 2, 4 and 5)
The reason this argument does not commit a composition fallacy is because, in this instance, the whole really is like its parts. Every contingent thing possibly fails to exist, and so the only plausible inference to make is that the series of contingent things could likewise possibly fail to exist. Nothing contingent has to exist. Therefore, the fact that anything at all contingent exists requires that its explanation be found in an external cause, which could only be a necessary cause.