Day 2 oita beppu
Day 3 oita yufuin
This arrangement is do-able as i had only visited Umi Jigoku(supposedly most scenic) in the morning.
https://www.japan-guide.com/e/e4702.html
After which i made my way to Yufuin to enjoy the rest of the afternoon till evening. The evening or so called "golden hour" will be great for picture taking.
I presumed that you have planned / booked accommodation at Yufuin for 2-nites.
In addition to
@zoossh 's recommendation of 1 hour bus ride, you may wish to check out Beppu --> Yufuin train schedules when you have arrived at Beppu station to decide how much time you wish to avail yourself at Beppu.
I took Yufu 3 train which definitely is much enjoyable while maximising the JR pass.
Day 6 nagasaki
Day 7 saga
Again, i would presume you would be based at Nagasaki for 2-nites to enjoy the night illumination at Inasa-yama and offerings of Saga.
From Nagasaki --> Hakata, you would be riding the Shinkansen and a limited express relay train to end destination of Hakata City.
Therefore the JR pass will proved itself to be useful and cost-effective while transferring with luggage across the platform seamlessly.
https://www.jrkyushu.co.jp/english/train/relay_kamome.html
*If you are traveling to Nagasaki, please make a transit from Limited express to
Nishi Kyushu Shinkansen at Takeo Onsen Station.
*If you transfer to the ""Nishi Kyushu Shinkansen KAMOME"" without exiting the ticket gate on the day of boarding, a single ticket will be issued. In this case, your reservations for the continual reserved seats (free) with the JR Kyushu Rail Pass will be counted together as one.
*""Relay KAMOME"" and ""Nishi Kyushu Shinkansen KAMOME"" are not covered by the JR KYUSHU RAIL PASS Online Booking service. If you are a JR Kyushu Rail Pass holder and would like to reserve a seat, please contact us to inquire at the ticket counter in a JR station after your arrival in Japan. (If you would like a non-reserved seat, you can ride with the JR Kyushu Rail Pass alone.)