I know that you can't have pictures of animated beings. But I read that if you cut off the head then it's fine, since the golden eagle only has a head and a pair of wings is it okay? Here's the Badge (in case you don't know how it looks like..)
According at Islamic Sharia it's not permissible to hang pictures of people or animals in house. The prophet PBUH says:
Angels do not enter a house in which there is a dog or a picture.
. لاَ تَدْخُلُ الْمَلاَئِكَةُ بَيْتًا فِيهِ كَلْبٌ وَلاَ صُورَة
Narrated by AlBukhary
We should erase everything that was in the image of a human or an animal being, it must obliterate only the face. In this case it's permissible to leave it.
You are correct. According to many scholars such as Sheikh Ibn Uthaymeen and others. Removing the head would remove the prohibition.
This was concluded from the hadith of Ali when he was sent to remove the head of any idol/image:
Narrated Abu Hayyaj al-Asadi: 'Ali said to me: I am sending you on the same mission as the Messenger of Allah (ﷺ) sent me that I should not leave a high grave without leveling it and an image without obliterating it.
So if you remove the head, You can hang the badge on the wall.