If it was me, I would make sure she is nesting in a nice, easily warmed area in her coop. Add more pine shavings around and under here when you either change out the eggs for what you want or when you catch/force her off her nest for a few. Maybe shift the nest into a cat carrier or such to help reduce any drafts from open doors. I did that in the summer but mostly to keep my hens sister from messing with her. Caught that crazy quasi-broody scattering Summers eggs everywhere with poor Summer puffed up and chirping in distress a foot away.
Garages aren't the best idea with animals, often as not they are just as cold as outside but the switch can add additional stress to the bird. You would still need to apply the stuff above though the only advantage I could think of would be adding a heat lamp somewhere but then if the power goes out you are back in a trouble area.
Option two is to keep her on the infertile eggs she has now and find someone hatching chicks locally for a swap out in a week or so. Or if you have an incubator use that and swap.
Personally, I would go for the first option. A lot less fuss.
