The door handle does need to be a separate mesh 
You can’t move part of a static mesh.
It’s ok if you don’t have a separate door handle, if they’re ‘knob’ type ( rather than a lever ), you can just put a slightly larger door handle over the existing one.
The handle becomes a component in the blueprint, and you attach it to the door ( in the component hierarchy ), so that when the door moves, the handle automatically goes with it.
Then, you have to animate the handle separately, in the same way you did the door.