Ask your "travel agency based here in the US".
You are not the first person to do such a trip
All airlines use TIMATIC (as does your travel agent)
http://www.iatatravelcentre.com/pass...quirements.htm
https://www.united.com/web/en-US/app...aspx?i=TIMATIC
Many people leave one country on one passport and enter another country on another passport. Very common.