Re: Is this a Mason's badge?
The double headed Eagle, and the 32 indicate that it is a Scottish Rite Mason. There are several different levels of Masonry, the first being the Blue Lodge, which goes through degrees 1-3. The final step of the Blue Lodge is a lot of work, hence the term, "given the third degree".
Steps 4-32 can be accomplished through either the Scottish Rite, or the York Rite. They really aren't all that challenging, and most all of it anymore is ceremonial. There is also a 33rd degree, which is an honorary title conferred upon those that have proven themselves worthy.
Nice find !