He has shelter - you said he's rugged. Of course other shelter would be preferable, but he's still better out than in, health-wise. If you must keep him in & don't have time for more yourself, can you get someone else to take him out another time or few per day? A little evening walk is better than nothing, but not adequate to keep him healthy. Also for his mental wellbeing, is he at least stalled with access to other horses?
As for his feet, if his feet are unhealthy & prone to bruising, using something like Vettec SoleGuard is one treatment. Unfortunately lack of exercise also means lack of hoof health, so I wouldn't expect much improvement ATM.
Re feeding, you can put his hay ration into a small holed or doubled hay net and this will keep him from gorging it quickly, keep him going for much longer.
Horses belong outside. Keeping them stalled for long periods of time stresses them mentally, physically and emotionally. Hooves deteriorate, gastric problems develop, osteoarthritis begins and horses become prone to ligament and tendon injuries, not to mention the various vices and temparment issues that can begin to arise..
Pack his hooves, or but them in easyboots, chuck a blanket on him and let him out during the day, no matter the weather.
My mid/upper level dressage horse is turned out with other horses in a large area where he can run and play all day long. No matter the weather he is out. The snow is up to his belly in some places and it dips down to -40 some days.
In most parts of North America, there is no excuse for keeping a horse stalled.