Ellen Sutton (born to Harriet and John Sutton 1868 in Wolverhampton) appears in the 1871 Census, but not the 1881 Census, even though her siblings do. The death index shows that a 4-year-old Ellen Sutton from Wolverhampton died in 1872, so this may be her, but there are later census records for an Ellen Sutton from Wolverhampton in a different household as a lodger and a servant, respectively. Was it common in the 1880s for a 13-year-old girl to leave home to work as a servant or live with relatives?
The Education Act of 1880 was the first to make education compulsory for English and Welsh children, but only between the ages of five and ten, and even then many parents did not send them to school as they needed the money - just as happens in many poor countries today. So, it is entirely possible, as you suggest, that a thirteen year old could have taken work away from home that required her to "live in" as a servant, or near enough to her work that she had lodge. My own grandmother left school at twelve, but she was lucky enough to be able to live within walking distance of her job. Her father was a butcher and they lived in a very small house in central York, Even they had a live-in servant, per the 1891 Census, who came from fifty miles away. This young girl was sixteen, but near neighbours employed a girl of fourteen, so your young lady of thirteen, away from home, is certainly not impossible. I hope this helps and doesn't make things even more difficult!