From author's comment:
SQLite has functionality that can be compiled in or out. For example loading extensions is off by default as is extended column metadata. Virtually all Linux distros compile SQLite with them turned on. APSW needs to know this at compile time and cannot defer to runtime.
You can have APSW automatically detect the system SQLite configuration. Create a file named setup.apsw alongside the extracted setup.py with these contents:
[build_ext] use_system_sqlite_config = True
The maintainers for Arch, Gentoo, and Debian all switched over to doing that a few releases ago making their lives easier.