Is it permissible to say "for the sake of the Prophet" or "for the sake of God"?

The phrase "min ajli" (من أجل) means "because of," and the saying "I ask you by Allah or by the Prophet" means "I ask you because of them." The letter ba (ب) in "bika" (بك - by you) can be for oath or for causation.

This phrase is a question, not an oath. It is a creature asking by Allah or by the Prophet, and there is no harm in a creature asking by the Prophet, peace and blessings be upon him.