Ok first of all I'm not a big fan of those liveries who dictate to owners if and when their horses are turned out. Horses are by nature outdoor animals and it is also considered cruel to lock them in. It can cause them stress and anxiety. I know in your case this can't be helped for reasons out of your control right now but that is my two cense on that subject. Maybe these other places might be worth considering. Some people will travel for the sake of their horses. Heck I moved to be closer to mine so I didn't have to travel so that they could be out 24/7. However, since we're not really discussing that at this moment in time I don't feel it is even up for a debate so I'll leave my thoughts on that there and say no more about it.
As for rug? I don't think she needs 200 g rug all the time, she doesn't look like the kind of horse that needs warmth on a constant basis. Only consider light weight if it's raining, medium if it's snowing. Do not cover if there is no bad weather and there is shelter as you mentioned? She honestly does not need it all the time. Let nature take it's course on a horse such as this. It is within her best interest.
As for ad lib hay? If she's kept in it is your responsability to provide her with as much hay as possible since she is kept in and thus limited on forage and if it is not possible for you to supply it when needed then maybe see if you can come to some sort of arrangement with people at your livery to do so? You should not keep a horse in for several hours without free choice hay. You say she is like a black hole? This could be due to lack of feed source for several hours at certain times of the day if you are not feeding her on a more constant basis. Horses, like most craetures, will gauge if they're not recieving the proper intake of food. They generally do not overeat if fed properly. This does not mean that you should give her large meals however. Little and often is best. Personally I'd advise not using hard feed but if it is required only small meals such as 300-500g per serving. Horses have small stumaches, around the size of a rugby ball. Despite their abiltiy to gauge they will not digest food properly through heavy eating and this can make them sick and also waste your feed due to possible diarrhea reaction. You are on the right track with haynets and net covered bales with small holes as this, as you said, will slow down her eating so she cannot gauge but she must have ad lib available to her on a more constant basis. Even if it means putting hay out 2-3 times a day it is required. So do this wherever possible, even if you can't put it out in her field do so in her stall.
If you feed her like this I can assure you she wouldn't need any hard feed, or very little of it depending on her nutrition intake. It is possible she might need some sort of supplement. I'd advise considering mineral blocks and plenty of water.
Either way I can certainly say she looks great in the photo. Nice glocey coat and good formation in her body. You should be proud of that at least