Have your white discs on the actual card snapped? Usually, it's the black fir trees pictured in Tim4x4's post above. Lots of my black fir trees have snapped. I wonder if you can get the white inserts that are bonded on to the door card?
I was unsuccessful in my hunt for the inserts you refer to, as a repair I replaced them with screws.
I will give a little more detail.
Having found at B&Q the shortest screws they had in which the head of the screw was able to locate in the fir tree slot just where the white inserts locate.
I then measured the core diameter of the screw (that's the centre portion not including the thread), this is the size of hole I am now going to drill into the back of the door card.
I am drilling this hole as deep as I can without breaking out the other side (this will ensure the screw remains unseen when the door card is fitted).
The hole size is allowing the self tapping screw to cut a thread into the door card material.
The screws I am grinding a little off the length of the thread because I do not want them to push through to the other side of the door card.
Now for fitting the screws into the door card I am using a small amount of adhesive (I used Locktite all purpose extra strong adhesive).
Screwing them into the door card until the fir tree will just slip on with a little pressure.
Give it a little while for the adhesive to set and Robert's your mothers brother.
I must have done this to around 15 of the fixings on the 4 door cards (not 15 on each card) and have taken the door cards off a number of times since it works perfectly and totally an invisible repair.