Upvote:2
The Catholic Church recognises the laws of the countries in which it is present. An information booklet issued by the Australian Catholic Marriage and Family Council for the Bishops Commission for Pastoral Life of the Australian Catholic Bishops Conference - 'Getting Married in the Catholic Church: Frequently Asked Questions' - contains the following paragraph:
As well as being a registered marriage celebrant in order to perform any form of marriage ceremony. the priest must include prescribed words in the marriage ceremony and must officially register the marriage as soon as practicable. There are serious penalties for non-compliance.
In any case, why would you want to have a marriage ceremony that is recognised only by the Church and not by society at large? If you wish to maintain financial independence either during marriage or in the event of marriage breakdown, then talk to a solicitor about managing your financial affairs.