Of course you should treat all internet advice as unreliable, but my understanding is, you DO NOT need a visa to transit CCU. You will of course have to spend all your time locked in the small transit area, the pleasures of which can be exhausted in about 2.5 minutes. If you are game, you should leave the airport and negotiate a taxi to drive you around for a few hours. But then you WILL need a visa, and you should not do it if this is your first time in India.
As stated above, Biman flights are via their capital and hub, Dhaka, and so should be avoided. (Of course BA also has nonstops to DAC and then you could reasonably take Biman to BKK.) I agree that Thai is better than IC, but I gather from your post that Thai does not have flights that day (they do this route 3 or 4 times a week). IC is not nearly as bad as it used to be although it is not yet a world-standard airline. Be patient.
Have you checked award availability to CCU? CCU awards are not that easy to get either, since BA is the only airline flying to CCU from anywhere in Europe, and that's only 3 times a week.