A three-city week is possible on a calendar, but each changeover divides a day between checkout, station or airport, luggage and the next room. Count the sightseeing hours that remain before deciding whether the island adds enough to the trip.
Give each stop a reason
Seoul needs the neighborhoods or fixed plans that brought you there. Busan needs more than an arrival evening if you want its coast or port districts. Jeju needs a chosen area and a transport plan. If one stop has no specific purpose, that is the first city to remove.
Protect the Jeju block
The island leg also has airport travel and a new hotel transfer. If you need to collect a rental car, include that handoff before setting the first outing. Keep the first Jeju evening near your base; do not count the flight date as a full sightseeing day.
Use the two-city version when the week is tight
Seoul and Busan fit more comfortably when you put KTX on a separate transfer date and give each city several consecutive nights. Seoul and Jeju can also work when the island has time for a planned route instead of an airport-to-photo turnaround. Check the actual flights before booking rooms.
When all three make sense
Keep all three only when the arrival and departure times leave useful days, the middle flights are already chosen and each stop has something distinct you both want. Otherwise, the third hotel and transport handoff can replace time you expected to spend in the destination.