Is it permissible to remove thick hair from under the eyes using threading, keeping in mind that the questioner has a beard?

The fundamental principle is to exempt the beard and not tamper with it. The definition of beard hair is that which grows on the cheeks and chin. Any hair other than that is not included in the definition of a beard, so there is no harm in removing it by any means, especially if its retention disfigures one's appearance or bothers the eyes.