I would reach out to a wildlife rehabilitator near you, preferably one who specializes in birds. They would be the best person to help.
For what it's worth, if it is the egg of a ground nesting bird, seems like either she's only just started a clutch or something happened to the rest. They don't sit on them til they have a full clutch. So if it's the former situation, she probably wouldn't have been incubating it yet. Though if you've already added heat to the equation, I think you need to keep doing so or the developing chick will die. Of
course, I'm no expert. Just a thought.
Please be gentle with yourself. Bird eggs are amazing and it's understandable you'd be excited. Given your description of the area it was in, it sounds like all kinds of worse fates could have befallen the egg. Who knows, maybe moving it will cause the mom to choose a new site somewhere less fraught with the perils of dogs, humans and mowers, thereby boosting the odds for her future eggs making it to adulthood?
“Action on behalf of life transforms. Because the relationship between self and the world is reciprocal, it is not a question of first getting enlightened or saved and then acting. As we work to heal the earth, the earth heals us.” ~ Robin Wall Kimmerer