Would it be a bad idea to put a couple (potentially) fertilized eggs under the broody OEGB?
Personally I would not. I'll explain.
Before a hen or pullet starts laying eggs she builds up excess fat, mainly in the pelvic region. I've butchered enough pullets, hens, cockerels, and roosters for this fat build-up to be obvious. This fat is there for the hen to live off of if she goes broody so she can stay on the nest and take care of the eggs instead of having to go out to forage a lot. It just doesn't make sense that Mother Nature would require a hen to starve herself to death and be in miserable physical shape when they hatch. They will lose weight, but it is fat put there for that purpose. Losing weight does not mean they are in any danger or suffering.
How long does that excess fat last? It will vary by hen. Some hens store more fat than others. All hens come off of the nest (if they are normal) to eat, drink, poop, and maybe dust bathe. Some come off quite a bit, like yours. Others hop off and hop back on pretty quickly. Typically a hen will break from being broody when that excess fat runs out. I arbitrarily use 5 weeks from when they go broody to when the eggs will hatch as my cut-off. Most can go longer. I've never had one go less but with living animals you do not get guarantees but the vast majority will break when that fat runs out.
The reason I would not give her eggs is that she has been broody long enough to surpass my arbitrary cut-off. She might stay broody long enough to hatch those eggs but I would not try.
I personally do not like just setting two eggs. Not every egg hatches. A lone chick with a broody hen will do OK until the hen weans it, but after that it has to make its own way with the flock. That can be challenging for a single chick as they are flock animals, they really want to be with others. Also, you don't know what sex you will get. I suggest you set as many eggs as you can handle when you set eggs so if they all hatch you can hopefully avoid the single chick issue.
I've given newly hatched chicks to broody hens several times. It practically always works but there is a catch. The chicks need to be as young as possible. The hen and chicks need to imprint on each other. The older they get the less likely the broody hen will imprint on them. The older they get the more likely the chicks are to have imprinted on something else and will not accept the hen. I've had tremendous success if the chicks are three days old or younger. I've had reasonable success if they are a bit older. But, guess what. You do not get guarantees with living animals, anything can happen. If you try you need to be prepared to brood then yourself.
The way I give chicks to a broody is to wait until it is dark and slip the chicks under her. Then I'm out there at daybreak to see how it is going in case I need to intervene.
If you break her now there is an excellent chance she will go broody again sometime. You don't know when and you don't know if it will be a convenient time, it may never happen. But I'd expect her to go broody again.
Before she starts laying again she needs to replace the excess fat she has used up while broody. That may take a while so be patient.
Good luck!
