There are translations of two different varieties of Bengali, Bangladeshi and Indian. Neither is complete, although there is an unpublished manuscript of the final volume of the Indian Bengali translation.
Indian Bengali
সর্বাধিপতি আংটি (Bengali [Indian]), purchased on-line from The Dogears Bookshop, Goa, India in 2021 (paperback)
Tolkien, J. R. R., সর্বাধিপতি আংটি, trans. by Aniruddha (Cinnamon Teal Publishing, 2012), 1, আংটির মৈত্রীসংঘ. ISBN: 978-93-81542-61-3
——, সর্বাধিপতি আংটি, trans. by Aniruddha (Cinnamon Teal Publishing, 2013), 2, দুই মিনার. ISBN: 978-93-80151-76-2
Only FR and TT have been published so far. RK exists in manuscript form, but has never been published. These volumes (including Hobbit) are print on demand so it takes a couple of weeks between ordering and shipping.
হবিট (Bengali [Indian]), purchased on-line from The Dogears Bookshop, Goa, India in 2022 (paperback)
Tolkien, J. R. R., হবিট, trans. by Aniruddha (Cinnamon Teal Publishing, 2011). ISBN: 978-93-80151-93-9
Bangladeshi Bengali
দি লর্ড অফ দ্য রিংস (Bengali [Bangladeshi]), purchased on-line from rokomari.com, Dhaka, Bangladesh in 2021 (hardback)
Tolkien, J. R. R., দি লর্ড অফ দ্য রিংস, trans. by Moheul Islam Mithu (Aishwarya Prakash, 2020), 1, দি ফেলোশিপ অফ দ্য রিং. ISBN: 978-92188-3-X
This is an unusual edition because "Lord of the Rings" is not translated into Bengali, but simply transliterated. This edition shares this feature with Urdu, Thai,and Indonesian.
Only FR has been published so far.
দ্য হবিট (Bengali [Bangladeshi] [H] A), purchased on-line from rokomari.com, Dhaka, Bangladesh in 2021 (hardback)
Tolkien, J. R. R., দ্য হবিট, trans. by Rahman (Adee Prokashon, 2015). ISBN: 978-984-91327-8-3