A CLA generally does not add clauses to the license. It generally contains extra rules contributors have to follow to contribute to the "main" codebase, but these are unrelated to the license. If a project has a CLA, you still have the full rights the license grants you, and you can do everything that license allows you to do without signing the CLA. The CLA only governs contributions to upstream, and I'm not aware of any open-source license that says anything about those.

As an example, many projects demand copyright assignment through a CLA from contributors, which does not add any restrictions to your use of the project allowed through the BSD license. You still can modify, redistribute, ... the source and products build on it as you wish, the CLA is only relevant if you ask the upstream project to merge some of your changes. The BSD license says nothing about how the upstream project has to accept your proposed changes.