Bangladesh cricket will be in serious trouble if it tries to confront the BCCI in this manner.
Any league has the right to ban a player if it believes that the player’s presence will not be well-received by spectators and could negatively impact viewership. It’s their league and their money—so ultimately, it’s their decision.
In-fact, on multiple occasion BCB itself didn't allow Bangladeshi Players like Taskin, Fizz and few others to take part in different international leagues including IPL and refused to provide NOC mandatory to take part in those leagues.
It's always up to the individual board. BCB believed it was in country's best interest not to grant permission to certain players to participate in overseas leagues, just as the BCCI decided it was better to organize IPL without FIZZ. At the end of the day, it all comes down to money and commercial success.
However, the way the Bangladesh Cricket Board is making a mountain out of a mole hill and reacting to this decision could easily end up digging its own grave.
It’s also important to remember that there will be occasions when Bangladesh is scheduled to host international tournaments on its own. If India refuses to play in such events, there’s a very high chance the tournament will be shifted elsewhere. The BCCI generates more revenue for the ICC than all other boards combined, and the ICC is unlikely to organize a major tournament without India’s participation.