I would not put her on the lawn if it was seeded as it usually contains ryegrass and clover. A small pony can at enough calories in one hour on grass for its daily needs. I leave a square next to anywhere they are grazing fenced off so you can see by the grass growth what they have eaten. Its sometimes hard to remember that there is no grass because they have eaten it!!
Have you heard on a paddock paradise, its where you make an inner fence round the edge of the turn out area and put food and water in different places, there fore they walk further also buy small whole hay nets and always put the forage in them so it takes longer to eat.
I would not feed hay, replace with good clean barley straw or if you can get it oat straw.
Pony dieting is a full time job.