Upvote:4
I opted for Satamisaki (佐多岬, Cape Sata), the southernmost point of Kyushu. The observation deck (展望台) is ~73 km from the launch area as the crow flies. And it seems to be a fairly popular destination for watching launches, as there were plenty of other people there too - some with telescopes and other equipment.
According to Google Maps, the parking lot closes at 5 PM, but it turns out that the parking area is open and free for use throughout the day and only the information centre and other buildings are closed. The toilets also remain open. Unfortunately, it seems that the only vending machines in the area are inside the buildings and the nearest convenience stores are 20+ km away, so stock up on refreshments before coming.
Other possible locations: