Although I really don't like individual turnout as a long-term thing and much prefer my horses to be out with others, I'm quite happy for mine to go out alone for a few hours as he finds stabling stressful and I like him to be in for as little time as possible. Ultimately, although some other owners are able to show up at the yard whenever they like throughout the day, I work full time and have to fit my visits to the yard around my job, so I turnout early on a morning, and bring in late afternoon.
Although Salsa will have a bit of a tantrum at first, usually if he's the last one out, he always settles down again. I did have some problems at a previous yard where another owner brought her horse in at 3pm and insisted on bringing Salsa in so he wasn't alone, which meant he was stressing out in his stable potentially for three hours if I had a late meeting. I was a bit baffled, as I didn't even know who she was, and was quite furious that someone was messing my horse around without my permission and getting him wound up. I finally tracked her down after a few days, as I managed to get up to the yard early. I almost had a heart attack, as he wasn't in the field and wasn't in his stable - the other owner had fetched him in and was standing at the back of the yard talking to someone!
I tried my hardest to be polite, as she was only trying to be considerate, but I had to tell her to leave him in the field, as I'd rather he was out by himself than in his stable weaving and stressing until I got there.
It sounds like your share horse is maybe one who is liable to stress if they're alone in the field, so there's not really much you can do about it, especially since the owner doesn't want him out alone. Maybe one way around it would be to offer to top up hay and water for some of his fieldmates on a morning if one of their owners could turn out for you when they're ready? Or see if someone wouldn't mind you turning out their horse with him (you never know, you might be giving someone a sleep in by offering
)
It's one of the reasons I now have two - they always have each other for company!