Problem adding mist to a 3D Element

Hello people.
I have a video as a background and I’m added a 3D Element far away. That’s fine.
The problem is when I try to add mist (I think it’s called atmospheric perspective) to the 3D Element to make it look like it’s far away. For that reason I need to add the mist only to the 3D Element, but not to the background.
To do that, I’ve added a mix node and dropped it between the render layer (that contains the 3D Element) and the alpha over node (which connects my 3D Element to the background). I’ve added a mist pass to the render layer and connected it to the factor of the mix node. That adds the mist to my 3D Element and also to the background.
How can I do it so it adds the mist only to the 3D Element?
I’ve tried to check to Convert Premul box in the Alpha Over node. That fixes my problem but creates a new issue: it draws a line around the 3D Element and looks terrible.
Any ideas of how to do it correctly?
Thank you very much.

You might well be able to achieve this effect “in post” by a combination of two factors: (1) “forced perspective,” then (2) reduced hue-and-saturation (HSV node). This might be just as effective – if not more so – than “mist.”

If an object is “far away,” it is of course smaller, and there’s a lot of atmosphere betwixt here and there. The object takes on a bluish hue, and its colors become desaturated.

I would manipulate the image in this way before alpha-over onto the video. I think that you can easily omit the need for mist. If you still want that, maybe blur or defocus. But I doubt you’ll need it. An object in the distance might very well be clear.

That worked perfectly.
Thank you very much!!