Is it permissible for a husband to divorce his wife if he saw in a dream that the Prophet informed him of his wife's collaboration with the Americans, and does this constitute betrayal that warrants divorce, especially given the lack of harmony in their interactions and the presence of two daughters on the verge of marriage?

Visions do not serve as a basis for legal rulings (ahkam shari'ah) or worldly dealings, due to their unreliability in transmission or interpretation. There is no legal evidence (daleel shar'i) that obligates acting upon them. There is no difference between seeing the Messenger of Allah, peace and blessings be upon him, or anyone else, due to the possibility of error in perception and the unreliability of the dreamer. The fundamental principle is that what the Prophet, peace and blessings be upon him, conveyed while awake takes precedence over what is conveyed in a dream. Divorce has its own specific rulings and is unrelated to these visions. Furthermore, it is proper to write the salutation upon the Messenger of Allah, peace and blessings be upon him, in full.