Also, if you really want to stay with cpan, you can (manually) install CPAN::SQLite and feed that from another machine with the updated index. But it's so much more convenient to use App::cpanminus instead ...

I remember having tried cpanm but I had issues.

I may try again, but my understanding is that the missing EXTERN.h will not be fixed by getting a cpan-client to work.

This does not come with a module but should part of the perl-installation (or am I wrong?).

No, CPAN::SQLite needs DBI and DBD::SQLite, and thus CPAN::SQLite is not part of the Perl core. And now that I'm writing about this, this approach is a non-starter for your case, because both DBI and DBD::SQLite will need a C compiler and the Perl headers, which you don't have (yet) ...