I don't think adding just one flake of alfalfa per day, and letting them hand-graze for an hour on someone's lawn will do the trick here. I'm in the camp of giving them free choice hay. I get that you're on a budget, but hay is cheaper than feed normally (I guess it varies by region). If you can't get your hands on good hay at a decent price, consider hay cubes (preferably soaked) as a substitute.
Also, do a fecal. It's not hard, you get a stool sample, drive it to your nearest vet lab, and they tell you if there are parasites. Around here, it costs about 20$. When I got a new horse and he colicked within a couple of weeks of being here, we did a fecal sample and he tested at over 400 (anything over 200 should be treated). I think that's why he colicked. I dewormed him immediately. I deworm all three of my horses twice a year systematically, even though the others rarely show any worm counts. You can't tell if a horse has worms just by looking at him.
I do like beet pulp a lot myself so consider that option too. Make sure you get the molasses-free version though.
And if you really can't feed more than twice a day, invest in slow-feeder hay nets with small holes to make it last longer. Horses need forage most of all - it's what they're made to eat, and it keeps them from getting ulcers by keeping their digestive system working at all times. When a horse goes without food for more than a few hours, the acid production in his stomach causes irritation, and eventually, can cause painful ulcers. Unlike us, they cannot shut off their acid production so if there is nothing in the stomach, that acid is just splashing around. I'd follow the recommendations of the folks here - 24 lbs of hay per day, per horse. That's about what I feed mine, along with some beet pulp and hay cubes, and mine are all 14.3 and under!!! 12 lbs is nothing.