CathInfo.com

If the woman prays and has no aversion to God, the problem is purely mental.  If she is antagonistic towards prayer or religion (even if she goes to Church), then there is a spiritual issue.  Sometimes there is both.  But i'd rather deal with a purely mental issue, rather than a spiritual one, as if demonic oppression is involved, there is no chance for anyone to help her except herself (free will), or a priest.

I'm not an expert but I listened to an in-depth podcast where an exorcist was talking about current events.  He was not Fr Ripperberger, but he's in close contact with him and other exorcists in the US.  They compare notes and statistics to keep up to date on trends.  This priest said that, in their collective opinion, the number of possessions are about the same as always, less than 5%.  But the number of demonic oppression cases are up sharply, around 20-25%.  Demonic oppression is where a person's sins have made them open to demonic attack.  The person isn't possessed, they still have free will, but they are attacked mentally and physically beyond normal temptations.

Don't know if this applies to the woman in question, but maybe it does.
 

OP here.

No diagnosis as far as I'm aware, unless perhaps when she was a minor. And a diagnosis is unlikely, as I seriously doubt either of them would entertain any psych professional (which I understand).

The behaviors in question include pretty intense gaslighting/manipulative behavior, including texting men at the chapel (including the priest) at the most unusual times with fairly grandiose claims about she has been victimized by them. The messaging is belligerent and antagonistic, and challenging. Her husband is aware of and dissatisfied with this behavior and does not seem able to influence it.

Strange, grandiose, attention-seeking claims are made regularly. For instance, she once told everyone her husband was dying. When a friend went to their home, he was treated as though nothing was going on. She tried to just make small talk (her husband was not dying).

Grandiose and exaggerated (to the point of basically being lied) claims about injury and illness are very common.

The family has moved. A lot.

How is this poor husband is going to convince this woman to change her nutrition habits. 

"Sweetheart, let's go grocery shopping together and I'll show you which foods will be healthier."
"I don't like those foods and I'm not going to eat them. You can cook that yourself if you want it."

"Darling, it would be good for the children too."
"Now you're accusing me of being a bad mother on top of trying to feed me yucky meals."

And so on. What's he supposed to do, take over her female responsibilities in the kitchen and then have to deal with her at the dinner table as though she's his oldest child? From what the OP says, she's got serious problems with mortal sin that aren't going to be fixed by changing her diet no matter how hard he tries that angle. The priest needs to guide this husband, not a dietician.