Pick big bang if the app can go read-only for the whole copy, phased if you can move data in slices, and trickle if you need near-zero downtime and the source can stream changes.
In our test, COPY loaded 500,000 rows at 735,000 rows per second against 17,500 for autocommit inserts, which can turn a 95-minute job into a 2-minute one at 100 million rows.
Prove the data arrived with a fingerprint of row count, a numeric sum, and min and max dates on both sides, and keep the source read-only for seven days as your rollback.