Why $A \cap B = \widehat{\widehat{A}\cup \widehat{B}}$ does not holds for the class of Recursively Enumerable Languages?
The claim "If some language class is closed under any two of the three operations namely, Union, Intersection &amp; Complement then it must be closed under the third." is not accurate.  I'm not sure where you got it from, but it's not correct.

Recursively enumerable languages are closed under union and intersection.  However, they're not closed under complement.  And, there's no way to express complement using just union and intersection.  So, the fact that they're closed under union and intersection does not imply that they're closed under complete.  There's no paradox/contradiction here.