You can’t change the license of the original code. But you can say “all changes from now on are licensed under the BSL”, with no need to physically separate the BSL and non-BSL parts. The result is a codebase where users have to comply with both the original license and the BSL. For most permissive licenses, that’s basically the same thing as a pure BSL codebase, except that typically the original license notice needs to be preserved in addition to the BSL one.

This isn’t the case if the original license is the GPL, because the GPL requires derivative works to also be licensed under the GPL.