First thing I thought was something to do with a bell clapper and I think it is a handle/weight for a rope/chain hanging from a bell. It's funny because I just got a hand me down bell 2 days ago and your piece made me think of it. I think your hand wraps around the eagle and the limb keeps your hand from slipping off. Similar arrangement to what SubT posted above.
Thanks very much for following up with pics and the measurements doc.
Now knowing the size, it looks as if the top loop has been soldered on.
At first I thought your piece was made of bronze, but seeing the copper verdigris leaching out, I'm thinking it's more likely to be made of pot metal.
"Pot metal is an alloy of low-melting point metals that manufacturers use to make fast, inexpensive castings. There is no metallurgical standard for pot metal. Common metals in pot metal include zinc, lead, copper, tin, magnesium, aluminum, iron, and cadmium. The primary advantage of pot metal is that it is quick and easy to cast. Because of its low melting temperature, it requires no sophisticated foundry equipment or specialized molds. In the late nineteenth century, pot metal referred specifically to a copper alloy that was primarily alloyed with lead. Mixtures of 67% copper with 29% lead and 4% antimony and another one of 80% copper with 20% lead were common formulations. The primary component of pot metal is zinc, but often the caster adds other metals to the mix to strengthen the cast part, improve flow of the molten metal, or to reduce cost."
I'm now thinking your 'eagle' was more likely designed to be a decorative piece as oppose to functional.