Why do you think it would 'do her good' to have a foal? Not sure what you mean by 'over produced'. I expect you know that many mares don't foal easily and many die and lose their foals each year. What if your mare dies and you have to bucket feed the foal every two hours, round the clock? What if the mare rejects the foal and it must be bucket fed? Who will do all this? I presume you have the money to pay for the stud fee for a quality stallion, but what about vet bills, if the mare has problems during foaling?
If you don't have facilities to foal her out and raise and train the foal, it would probably cost you a lot, to have someone else, do it properly. And 'properly' is the word to remember here. Few would want the responsibility, to be honest.
What would be your plans for the foal? Certainly your mare is not planning to have a foal in the future. She won't care if she's ever bred. It will be you who is bringing a new life into the world, AND responsible for training and working with it. If the foal is well handled, well started etc., and of course with superior conformation, you'll possibly find a buyer. But maybe not. There are tons of quality horses with excellent pedigrees, being sent to auction, each and every day, in these difficult times. The majority end up being sold to the meat man and shipped to Mexico.
These days, anyone considering breeding, should think long and hard about the decision. Good homes and buyers are just not out there, as a few years ago.