Does casting a card and paying for it work like this?
Due to the Oracle revision to this card, what your are describing will not work. The current errata states that you can only use  Lion&#39;s Eye Diamond's ability as an instant. The only abilities you can activate while a spell is being placed on the stack ("Pay Costs") are mana abilities such as tapping lands.

If Lion's Eye Diamond was not changed to specifically state that its ability is played as an instant, what you are describing would work. 

Relevant Oracle text:


  Sacrifice Lion's Eye Diamond, Discard your hand: Add three mana of any one color to your mana pool. Activate this ability only any time you could cast an instant.