The Lisbon treaty, ill-fated European Union constitution's

Understanding the author's intended meaning can make the elimination process easier.
The author's intended meaning is as follows: The Lisbon treaty followed the European Union constitution and the European Union constitution itself followed the Maastricht, Amsterdam and Nice treaties.

Between A & the rest:
If you get rid of the two modifiers, you have a simple sentence to begin with: "The Lisbon treaty would create a permanent club (EU's) rulebook".
Since the focus of the sentence is on the Lisbon Treaty not the constitution or the other treaties, you should bring "successor" closer to the Lisbon Treaty to describe it in the beginning of the first modifier.
Otherwise, it is not clear if "ill-fated" describes the Lisbon Treaty - the successor or the EU constitution, as in the case of (A). So (A) is wrong.

Between C & B,D, and E:
The presence of the word "in turn" suggests that the author wants to describe "the ill-fated European Union constitution" itself by the following modifier.

(B),(D), and (E) use the correct pronoun "which" to connect the noun closest to it, "the ill-fated European Union constitution" with its description "It followed or succeeded the three older treaties".
It is grammatically correct to use "which" since these treaties are unique and the additional pieces of information contained in the modifiers are non-essential.

(C) uses the wrong pronoun "that", which mistakenly refers to the Lisbon Treaty, implying that the Lisbon Treaty directly followed the older treaties, with nothing in between. So (C) is out.

Between B & D, and E:
(E) is clearly wrong for using singular form of the word "treaty" to refer to three treaties.
(D) is wrong for redundant use of "treaty".
(B) is more efficient than (D) for its use of "treaties".