While the Commons Clause is not an Open Source license, the BSD license allows you to do this.
To be clear, you cannot change the license of other people's code. You can however add your own code and license it under whatever terms you want. The resulting code then consists of
- your modifications, under whatever license you want
- parts of the original code, under BSD-3-clause
Since you are a license-taker for the BSD-3-clause material, you must still comply with its license such as providing any recipients with the copyright + license notice.
When people receive the software with your modifications, they must comply with both your chosen license terms for your modifications, and with the BSD-3-clause license for the other parts. For example, you can use the Commons Clause to prevent certain competing uses of the software as a whole. But since you will presumably provide the source code of the software, other people are free to use the BSD-3-clause code under the terms of that license, ignoring your restrictions.
If you provide binaries then the resulting binary would not be covered by the BSD-3-clause license, but you would still have to provide the copyright and license notice for the original code as a kind of attribution.