Now, the original wording of section II of article 27 (which dates back to 1917) was:
http://www.juridicas.unam.mx/infjur/leg/conshist/pdf/1917.pdf
II.- Las asociaciones religiosas denominadas iglesias, cualquiera que sea su credo, no podrán en ningún caso tener capacidad para adquirir, poseer o administrar bienes raíces, ni capitales impuestos sobre ellos; los que tuvieren actualmente, por sí o por interpósita persona entrarán al dominio de la Nación, concediéndose acción popular para denunciar los bienes que se hallaren en tal caso. La prueba de presunciones será bastante para declarar fundada la denuncia. Los templos destinados al culto público son de la propiedad de la Nación, representada por el Gobierno Federal, quien determinará los que deben continuar destinados a su objeto. Los obispados, casas curales, seminarios, asilos o colegios de asociaciones religiosas, conventos o cualquier otro edificio que hubiere sido construido o destinado a la administración, propaganda o enseñanza de un culto religioso, pasarán desde luego, de pleno derecho, al dominio directo de la Nación, para destinarse exclusivamente a los servicios públicos de la Federación o de los Estados en sus respectivas jurisdicciones. Los templos que en lo sucesivo se erigieren para el culto público, serán propiedad de la Nación.
Translation:
Religious institutions known as churches, regardless of creed, may in no case acquire, hold, or manage real property or hold mortgages thereon; such property held at present either directly or through an intermediary shall revert to the Nation, any person whosoever being authorized to denounce any property so held. Presumptive evidence shall be sufficient to declare the denunciation well founded. Places of public worship are the property of the Nation, as represented by the Federal Government, which shall determine which of them may continue to be devoted to their present purposes. Bishoprics, rectories, seminaries, asylums, and schools belonging to religious orders, convents, or any other buildings built or intended for the administration, propagation, or teaching of a religious creed shall at once become the property of the Nation by inherent right, to be used exclusively for the public services of the Federal or State Governments, within their respective jurisdictions. All places of public worship hereafter erected shall be the property of the Nation.
So, as you can see 1917's Constitution clearly states that churches in Mx could not own property. This law was in force when the organization obtained it's first registration back in 1932. Nine years later in 1943, they somehow "noticed" this tiny little detail and decided to restructure itself as a "civil association" which aim was SUPPOSEDLY no longer religious but "cultural" (as they have already told you, this part of the story can be barely found in pages 211-212 of 1995's yearbook).
An interesting fact you won't find anywhere else regarding this period (1943 - 1993) is that the only publication that was actually prepared and "adapted" for Mexico was Our Kingdom Ministry which kept its old title and at first, did not use the names of Jehovah or Jesus. They were known as the "Maestro Principal" and "Gran Maestro" respectively. Great emphasis was put on the adult literacy campaign and to quote the Bible a code using numbers was followed (for instance Matthew 23:10 appeared in the Bulletin as 40, 23:10).
Though they do admit that their decision had to do with the ownership of property, they try to include preaching and holding public assemblies as two other reasons which have no real basis to be credited.
The fact that this "cultural status" had more to do with "property ownership" is highlighted if you consider that after the reform of article 27 in 1992, on April 13, 1993 they immediately embraced back their "religious status".
Do they still report and announce with a great fanfare their literacy campaign achievements? Rhetorical question...