What is the legitimate Islamic solution for disputes between spouses, and is the husband entitled to refuse divorce until he fully guarantees his rights to his son?

If the wife stipulated residency in her country before marriage, then her condition must be met. Otherwise, she must obey her husband in moving with him wherever he wishes. If she refuses to do so, she is considered disobedient (nashiz), and the husband has the right to refuse divorcing her until she waives some of her rights.

As for the custody of the child in the event of divorce, the father has priority in custody if the parents differ on the country of residence, because the father is responsible for disciplining his son and preserving his lineage.

It is advised to reach an understanding with the wife and clarify what Allah has obligated upon her in terms of obedience. If she does not respond, then the Shar'i courts, Islamic centers, or trusted scholars are the ones who rule on these matters.