FK to a different database

Hey guys. Sorry for writing in English, but I need my boss to be able to read it as well.

Im working in a company where we are trying to create an online community for pregnant women of Colombia. Women will, the majority of times, have a doctor, so we need a foreign key from the mothers table to the doctors to avoid redundancy. However. The doctors will not allow us to copy their table and put it into ours, which would be the best solution, and we cant put our data inside their database because we are expecting some serious activity.
Since we are expecting a lot of calls requesting data from their base, it would be silly to call across databases, though it probably could be done with a trigger.

What would you suggest?

Can it even be done without a trigger?(at this point the language is not a problem)

What exactly are the backside of using a trigger?

Thanks in advance
Jakob