Very possible he's at least anaemic and very likely deficient in other nutrients even though he eats red meat. He needs vitamin c to absorb and only eating one vegetable is ludicrous! Most veg is GF but as he doesn't even have a diagnosis he's being an idiot!
Not least because something more serious could be going on with his digestive system.
How old is he? Changes in bowel habits etc can be indicative of several conditions aside from celiac disease which is actually relatively uncommon.
My ex when I met him would only eat 6 things, literally 6 and that included bread and potatoes!
I'm veggie and he reacted poorly to that and then I pointed out I ate more healthily and a far wider range of foods than he did which he had to accept was true.
It was his mothers fault (she's actually a lovely lady but failed on this aspect) as he was the youngest of 4 and when he went through his perfectly normal but shouldn't be conceded to fussy phase as a toddler she placated and capitulated. I think she's a bit too soft but also with 4 others to feed and run around after she was knackered! And so found it easier to just only give him what he would eat but she NEVER addressed it as he got older and he never moved on from that phase until I met him.
We had a LOT of discussion about it and what I did was I gradually introduced him to other foods (I started with sweeter foods like fruit and sweeter veg) not only eating but handling prepping and cooking (he could barely cook when I met him either, but he'd never had to - mums to army he'd never lived properly alone) and over time he became much better.
When I met him his stomach was a mess! He was always either constipated or had diarrhoea and always kept in a stock of laxatives and otc diarrhoea treatments. He'd even have regular spells of vomiting too maybe once every few months? He was also prone to colds and similar infections.
Actually thinking about it the first thing I did was get him to take a multivitamin with iron and start drinking a glass of fruit juice in the mess with breakfast. (He'd only have tea and toast for breakfast and only butter on the toast)
He was extremely slim because he wasn't having enough calories and was doing a manual physically demanding job and went to the gym every day.
This was pre internet times so I even found magazine articles and books from the library to educate him on nutrition and exercise. His PT instructor was introduced to me on a work night out and he was like "thank god someone is finally getting through to him about eating well" turns out he'd been trying to get through to him for ages and was actually really concerned about his health.
His colour gradually improved and he started gaining weight - which pissed him off at first but thankfully the PT guy pointed out to him that he'd been borderline underweight previously and he NEEDED to gain some weight and it was still mostly muscle anyway. He also pointed out that he was doing better on his fitness evals
It's a long haul thing, if he's a new partner to you it may not be worth your while but if it's an established relationship and you want to stay with him then it can be sorted. BUT he has to be willing to do the work, even if it makes him anxious right now. If he's willing to make a start then good but if he absolutely won't even consider discussing it then I would be out.
Agree too with those saying to get him to see a dietician (NOT a nutritionist. Anyone can be a nutritionist but dieticians are registered hcps with medical training) but he really ought to see a dr too AND be honest with them about the symptoms at the start of all this as more serious conditions MUST be ruled out first - this is IMPORTANT.
And if it is an ED he needs proper professional help with that too, ED's can cause major physical ill health as well as the mh distress.
I'd be very worried about his physical health given what we now know about the consumption of red meat and especially fatty meats.