There are several docklets that are user made on Wincustomize you could try out.

Here are some of the icons and docklets I am refering to:

https://www.wincustomize.com/skins.aspx?libid=29&search=mail

I'm sure other users have a recomendation or three.

Well a good deal of them are just icons. You need a mail docklet for Object Dock. Here is an example of a docklet:

https://www.wincustomize.com/skins.aspx?skinid=3958&libid=29

You would install something like this to the Object dock directory:  C:\Program Files\Stardock\ObjectDock\Docklets

Then right click on the dock and go to settings. Click on the Docklet tab and add the desired docklet. I'm certain a good deal of tweaking and setting adjustment will have to take place for your individual settings on your computer and email accounts.