Luggage recheck at Beijing Capital

Hello, I am traveling from Seattle, USA to Dhaka, Bangladesh via Beijing Capital. I'll fly from Seattle to Beijing with Hainan airlines and Beijing to Dhaka with China Eastern airlines. Both airlines say that I need to recheck-in my luggage at Beijing with China Eastern.

Is this something easy to do and can be done without a transit visa? I am not sure if both flights will leave from the same terminal. My stopover time at Beijing Capital airport is less than 6 hours, so I should not need a visa. Are there any unusual difficulties in claiming the luggage and rechecking in? I am traveling with Bangladeshi passport and USA green card.

Your response is appreciated.