Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Exeter to Osaka as you normally would.
For a flight from Exeter to Osaka, you will be flying from Exeter. Exeter (EXT) is just 5.3 mi from the center of the city. Exeter is the only Exeter airport and it handles 0 outbound flights to Osaka per day. 0 airlines have flights to Osaka from Exeter on a regular basis.