Can you configure the Copy Master/Detail Records button?

All fields are copied.  No choice.  And unless you are some kind of javascript brainiac, there is no way to replicate what it does in terms of copying children and grandchildren.

So, what I do is to call the COPY button Step 1 of 2.   I then key off the copy of field to detect that this is a copy and contrive the form to force the user to click a 2nd URL formula button which clears fields (including the "Copy of" indicator field and any other edits required.