Since a BREAK Pokemon "retains the attacks, Abilities, Weakness, Resistance, and Retreat Cost of its previous Evolution" can a Mew EX with the "Versatile" Ability use an attack of the Pokemon that the BREAK Pokemon evolved from (providing it meets energy requirements, etc.)?

Yes, it can. The underlying Pokemon's attacks are part of the BREAK Evolved Pokemon and are therefore able to be copied by Versatile.

Relevant cards