Yes, definitely add SQLite to your list of supported databases, and bundle a SQLite database for testing.
And don't just test SQLite if Oracle/PostgreSQL can't be tested. Test it always. Add a test_requires dependency on DBD::SQLite. If I happen to have Oracle and PostgreSQL up and running, and the appropriate environment variables set, then I'd want the test suite to run on all three databases.
perl -E'sub Monkey::do{say$_,for@_,do{($monkey=[caller(0)]->[3])=~s{::}{ }and$monkey}}"Monkey say"->Monkey::do'