How can I add a custom "Date" field as a field in a view?
I'm not sure why you're having trouble.  I just did this and it worked.  Here's what I did.


Add a "Date" field (widget: select list) to Content Type A.
Create a view that shows items of Content Type A.
Click Add next to Fields and search for the label for my "Date" field.


You should try looking through/searching your list of fields again.  It should be there.  (Even when a view is filtered to a specific content type, the Views module will often allow you to select fields that are not associated with that content type.)