The Church’s position on this is no different than any other “living-in-sin” position. No one is barred from participating in the mass, Catholic or non, but to partake in Communion you must do an examination of conscience and if you are practicing grave sin (mortal sin such as adultery, illegal drug use, theft etc.) you can not partake in receiving the Body of Christ because the Body of Christ can not become one with a person who knowingly and unapologetically remains in grave sin. Once forgiveness is obtained through confession, you can again receive Communion.
I forgot to add that you must be “catholic” to receive communion in a Catholic church.
I say that particular Church's integritysucks.
The answer should be to tell the wanna be Catholic to contemplate the importance of 1 Corinthians 5:11-13 and quit calling yourself a Catholic until you openly repent.