@sybbie719 has explained it in a nutshell. Limited funds means just that. There is not enough money to find everyone that Barnard would like to fund. You are in the category that is not need blind for admissions as an international student. So those limited funds apply to you.
If Barnard is your first choice school, go ahead and apply ED. IMO,it is likely you will not be accepted with an inadequate package, given what you have said about your family finances. Since you cannot get a student visa to even get into the country without showing specifically how you will pay for all of your transportation, education and living expenses, there is no sense in accepting you if you can’t afford to come here. You would have to be worth Barnard shelling out more than $300k over 4 years for you to be accepted.