Can I respond to a draw from Gitaxan probe?
If you want to activate Lions Eye Diamond before drawing a card, you must do so before Gitaxian Probe resolves. 

You can respond to another event, like your opponent playing a spell or activating an ability.

So no, there is no opportunity to respond to your spell once it has begun resolving. You will get priority after the spell resolves, and at that time you can activate your Lion's Eye Diamond.

See this question for more details:  Can a player respond while a spell is resolving?