I can't find any itinerary that would not require a visa to transit some country. I suspect no such itinerary currently exists, especially with many flights cancelled due to COVID-19.
Your proposed itinerary requires a visa from Cuba. Yours is one of the few nationalities for which Cuba requires a full visa and not just an inexpensive tourist card, and the itinerary also requires passing immigration.
Consider getting a C-1 transit visa from the USA. This is generally easier to obtain than a tourist/business visa. It will also let you exit the airport during your layover. And most of the cheap flights for DAC-POS will transit the USA. Having such a visa can also help you build a travel history and possibly make it easier to obtain other visas later.
You might also consider getting a DATV (direct airside transit visa) from the UK. You can then transit the UK and fly via the UK and Barbados, as Barbados does not require visas for Bangladeshi citizens. Depending on the day of the week you fly, this may be the cheapest option. But the DATV will not allow for exiting the airside transit area.