I think the problem is that your question is theoretical, and that often poses problems (for example, Matthew 22:23-33).
First, while the Bible does view a sexual union as becoming “one flesh” (1 Corinthians 6:15-16) and the Old Testament seems to have viewed sexual union as consummating marriage, not having sexual relations is not, in my opinion, basis for an annulment, as the Catholic Church might do, and as civil law may allow.
I would wonder if this couple understood anything about God’s word regarding marriage — texts like Ephesians 5:22-33 — or about love (1 Corinthians 13). When God brought Adam and Eve together, He said, “The two shall become one flesh.” It seems to me that a married couple is required by God to have sexual relations (1 Corinthians 7:1-5), and that both are being disobedient by refusing to do so.
I am uneasy with the way you have used a technicality (so to speak — the sexual union) to justify the setting aside of the marriage. Technically, the Catholic Church and civil government may nullify a marriage union that has not been consummated, but our Lord seems to strongly reject efforts to avoid our responsibilities, based upon technicalities.
The lawyer of Luke 10:15-37 wanted to avoid his duty to look out for his neighbor, based upon a technical discussion/debate over who his neighbor was.
Evil children sought to avoid their duties to their parents, based upon the technicality of a devoted gift (Mark 7:1-13).
And the Jews seemed to be using certain technicalities to avoid keeping their vows (Matthew 23:16-22).
It would seem to me that a person who has committed before God to live with someone as husband or wife should keep that commitment, rather than to seek excuses for escape, through a technicality.
When Jesus was asked about divorce by the Jews, He took a harder stand against it than anyone. Even His disciples were amazed. Jesus simply said, in effect, “It was not this way in the beginning. What God joined together, let not man separate.” I would take it that the reference to God putting together was not specifically a reference to the sexual act, but His bringing them together to be husband and wife.
If this couple was married in a church, then I would assume that they made vows to each other and to God, and the Bible is very clear about keeping our vows (Deuteronomy 23:21-22; Psalm 15:4; 76:11). Jesus said our “Yes” was to be “Yes,” and our “NO,” “No.” People who vow to be husband and wife for the rest of their lives need to keep their word.
The Bible has much to say about being kind and forgiving one another and putting up with one another. This couple should start doing that immediately and set aside their petty argument.
The second thing I would say is that your logic breaks down a bit when you say that the covenant of marriage is sealed by the sexual act. No text of Scripture really says this; it is an assumption, that could be challenged, even though the sexual union makes a man and a woman “one flesh.” By the way, if a man has been immoral and has had sex with many women, are these women all his wives, because they became “one flesh”?)
Malachi speaks of marriage as a covenant (2:14) long before Christ and the New Covenant.
I think it is right to say that a marriage is consummated by the sexual act. I think it is going a bit too far to say that if a marriage is not consummated by sex (for some reason, as you have indicated) that it is not a marriage at all. The couple still committed to live together as husband and wife for the rest of their lives, and should keep that promise, including the obligation to have sex, if they are physically able.
Joseph and Mary lived together as husband and wife, but did not have a sexual relationship until after the birth of our Lord. I don’t dispute that they were married, yet without sex.
One more thing (based a bit more on background information about the Old Testament times). It appears that once one became engaged, the only way this engagement could be broken was by divorce (as we see in the case of Joseph, when he learned Mary was pregnant). There was no sex, and only an engagement, but the relationship could only be broken as though it were a marriage.