I know this thread is exactly hot anymore but I do believe there was another aspect to the break-up.
Frederik's younger brother prince Knud was believed to be somewhat dim-witted. Whether or not this was justified I cannot say. But the fact is that he had suffer a certain amount of public rudicule; the chorus of a song from the time went "let's go over it it one more time for Prince Knud".
Now I know this is pure speculation but in latter part of the 19th century and the beginning of the 20th Eugenics was a widely recognized science (it's wasn't until later when the Nazis took it too extremes that it was abandoned). It is not impossible that CP Frederik and/or Olga had heard about eugenics (as in genetic hygiene not nazism).
So with a dopey brother the prospect of marrying a woman who is your cousin twice* over may have caused the CP to have doubts about the health of the their children and the future of the monarchy. Or maybe it was Olga who feared for the sanity of her children if they married.
I have to stress that this is pure speculation but I believe that this thought must at some point crossed their minds and it may have been a contributing factor in breaking the engagement.
* they both decend from King Christian IX & Queen Louise and Grand Duke Frederik Franz II of Mecklenburg-Schwerin & Augusta of Reuss Köstritz