Nellie, I'm a little perplexed by your comment that Princess Alexandra didn't have the rank to wear a full coronet. She is the granddaughter of a sovereign.
Both Princesses Margaret and Patricia wore the coronet of a sovereign's granddaughter at the coronations they attended. Princess Alexandra is of similar rank, so logically should be entitled to the same coronet.
Maybe I'm not understanding your comment correctly.
My guess is that she was considered too young to wear a full coronet, and a child's coronet was decided, using the precedent created by Princesses Elizabeth and Margaret.
I personally don't think it was a question of cost. Surely, there would have been some grandchildren of sovereign coronets in storage somewhere. Queen Victoria had so many granddaughters, I'm sure there would have been an unused one that could have been borrowed. That would have been cheaper than creating a new one for Alexandra (the difference in the cross pates does seem to indicate that it was a new one).