I was thinking it might constitute fraud on the part of the priest, since he normally acts as an agent of both civil and religious marriage. While those functions may be separable from a canonical point of view, I don’t know if the state agrees.
Further a priest might be reluctant to perform the ceremony, even if legal, if he realized that the couple intends to remain civilly single and not married. It still sounds like the couple wants to assume the rights but not the responsibilities of marriage.
When they go to file their tax returns, they will file as single. If they apply for a mortgage, they will apply as single, unless perhaps it is advantageous to apply as married. When they register in a parish, they will register as married. If one is in the hospital, will they indicate they are single or married? The whole thing just seems rather schizophrenic to me. “We are married but we aren’t, depending on what you want from us.”