How to remove date from Calendar.app dock tile?
Disclaimer: This is an untested (edit: now tested by OP) modification of the Calendar.app, you may do it at your on risk. Back up your computer to an external drive first.

Doing the following will stop the Calendar dock icon from having that date overlay.


If we right-click on Calendar in the Applications folder, an option comes up that says "Show Package Contents". That will reveal some (not all) of the inner workings of the Calendar app.
If you navigate to _CodeSignature and then CodeResources, you will find some of the resources the Calendar app uses to operate. Searching for dock inside that file reveals data keys including
&lt;key&gt;Resources/Calendar.docktileplugin/Contents/_CodeSignature/CodeResources&lt;/key&gt;. This points us to a file that may well control the Calendar icon (notice Calendar.docktileplugin).
So let's go there. Follow the file tree: back up one step, go to resources, and find the Calendar.docktileplugin folder. Deleting this folder will stop the Calendar app from automatically putting the current date on top of your calendar icon (Edit you might want to save a backup of this plugin as I mention in Edit 2 below). 


There you have what I believe is the solution to your problem. 

Edit: If the Calendar app crashes on launch, try deleting the original key (from step 2). If that doesn't work, my solution might not be so good.

Edit 2: Save a backup of the Calendar.docktileplugin folder (like maybe on your Desktop) before deleting it and remember where it went so you can always put it back if the Calendar app just crashes or if you want to restore the Calendar icon to original anytime.

Let me know if this helps.