I've changed ./Modules/_sqlite/util.h

//#if SQLITE_VERSION_NUMBER >= 3007014
#if SQLITE_VERSION_NUMBER >= 3107014 // Change
#define SQLITE3_CLOSE sqlite3_close_v2
#else
#define SQLITE3_CLOSE sqlite3_close
#endif

#endif

And tried to recompile with a new error:
*** WARNING: renaming "_sqlite3" since importing it failed: build/lib.linux-x86_64-3.8/_sqlite3.cpython-38-x86_64-linux-gnu.so: undefined symbol: sqlite3_errstr                                                                                                                                                                                                                                                                                                                          The following modules found by detect_modules() in setup.py, have been                                                                                                                                                                       built by the Makefile instead, as configured by the Setup files:                                                                                                                                                                             _abc                  atexit                pwd
time


Failed to build these modules:
_uuid


Following modules built successfully but were removed because they could not be imported:
_sqlite3

Went into ./Modules/_sqlite/connection.c and also changed:
//#if SQLITE_VERSION_NUMBER > 3007015
#if SQLITE_VERSION_NUMBER > 3107015 // Change                                                                                                                                                                                                                      PyErr_SetString(pysqlite_OperationalError, sqlite3_errstr(rc));

With these changes the build was successful and I can import sqlite3. But I have no idea if this has any unforeseen consequences.