Can one travel from Metro Manila to somewhere else in Philippines outside Metro Manila via another country during the COVID-19 Metro Manila lockdown? Or is it against the Metro Manila lockdown rules and would lead to being refused entry?
Technically possible, but in practice very difficult unless you're a citizen.
While some international flights to/from the Philippines continue to operate, there have been massive capacity reductions, both to Manila and other destinations.
More importantly, the Philippines have also banned foreign visitors from countries with "local coronavirus transmission", an ill-defined group that is likely to cover virtually every larger country. This doesn't apply to Philippine nationals and residents though: http://www.cnnphilippines.com/news/2020/3/12/Philippines-expands-travel-ban-to-all-countries-with-local-COVID-19-transmission.html