How can the problem of the head covering be solved with the fiancée, who refuses to wear it, given that the questioner wishes to marry her and sees that his friends' wives wear it?

The covering of the face is a matter of scholarly disagreement among jurists, with the preponderant opinion being that it is obligatory.

If the girl is a student of knowledge and her preferred opinion is that it is not obligatory, or if she is an ordinary person acting upon the fatwa of someone she trusts, then there is no blame upon her for uncovering it. However, if her husband commands her to cover it, then obedience to him becomes binding upon her.

If you are unable to convince her to cover her face, and you fear that this will be a cause for future dispute, then it is better for them to separate.