There is no direct train from Kolkata to Feni. There is a five times a week service from Kolkata to Dhaka Cantonment Leaves Kolkata around 8am, arriving in Dhaka around 4:30pm. You have to make it from there to Feni by Bangladesh Railway service, four trains a day taking between 4 and 6 hours. There are late evening departures arriving middle of the night or departures during the day arriving later in the day.
From Dhaka there is rail service upto Chattogram (erstwhile Chitagong) at present. Extension to Cox's Bazar is scheduled to go on line in June 2023 according to this article:
Government plans to bring every district under rail connectivity, Railways Minister Md Nurul Islam Sujan says
Until then there is very good bus service both from Dhaka and Chattogram to Cox's Bazar.There are flights too from Dhaka to both Chattogram and Cox's Bazar (about one hour flying time) both Biman Bangladesh and US Bangal Airlines.
If my main goal was to visit Feni and Cox's Bazar I would not fly to India. I would fly directly to Dhaka with an airport much larger and with more international traffic than the one in Kolkata. I would fly Emirates or Qatar Airlines directly to Dhaka with a connection through Dubai or Doha. Qatar through Doha has the additional advantage of getting US C&I pre-clearance at Doha on the way back.
If visiting India too was a goal then of course flying into India makes sense. Get the latest information from both Bangladesh and Indian Embassies on whether they will do Visa on Arrival through the land crossing or you will need to get Visa from the Consulates before travel, and if the latter whether you can get them from Consulates in the US or they will issue those only in Bangladesh and India. These rules change frequently and it is best to get informed well before hand.
Hope you find this information useful....
BTW I was born in Kolkata and still own significant inherited property there and in its suburbs jointly with a sibling and a whole bunch and cousins. My maternal Grandfather's family came originally from Borishal (Barisal) District, now in Bangladesh, and on the verge of getting rail service for the first time ever! They moved to Kolkata a generation before the partition.