Does the wife's distress, boredom, headache, her feeling of something moving inside her body and eyes when reciting the Quran, the deterioration of her condition after ruqyah, and then her partial improvement, necessitate a referral regarding her treatment?

You should turn to Allah by supplicating for her recovery, and persist in dhikr (remembrance of Allah), acts of obedience, and reciting the Quran, especially Surah Al-Baqarah and the Mu'awwidhat (Surahs Al-Falaq and An-Nas). There is no harm in seeking Ruqyah (Islamic exorcism/incantation) from righteous people, for it benefits, by Allah's permission, if its conditions are met and its impediments are absent. What she feels after Ruqyah might be from the treachery of Satan, so beware of despair and resorting to charlatans. It is also advised to take her to specialized doctors, as her illness might be organic.