Is it Church approved?
Yes. The local ordinary, His Eminence Cardinal Péter Erdö, Primate of Hungary, established The Flame of Love Movement as a Private Association of the Faithful with Juridic Personality. That last phrase is important. Private Associations are not necessarily approved but any Association with Juridic Personality must be approved by a competent ecclesiastical authority. So though the Movement was established in Budapest, the approval is for the Universal Church.
Canon 322 – § 1 . A private association of the Christian faithful can acquire juridic personality through a formal decree of the competent ecclesiastical authority mentioned in can. 312. §2. No private association of the Christian faithful can acquire juridic personality unless the ecclesiastical authority mentioned in can. 312, § 1 has approved its statutes.