There were many families who went to France for work and had one or two of their children born there. Nothing unusual about it. Also it would have been highly unusual for the French to allow an English family to adopt a French child.
In the early 1820's Aaron Manby took over the running of the Charenton Ironworks near Paris. He was an ironmaster from Birmingham who had fabricated an iron steamship at his Horseley Ironworks in 1822. The parts were then sent down the Thames to Rotherhithe to be assembled. The steamship was named Aaron and made it's début on the Thames in June 1822. A few days later it sailed to Paris. The Charenton Ironworks were so successful that another was opened at Chaillot.
At the time the English were superior to the French in iron-working so Manby recruited several hundred English workmen to come to France with the promise of good wages.
According to the book "The Industrial Revolution In Iron: The Impact Of British Coal Technology In Nineteenth-Century Europe" by Chris Evans and Göran Rydén, a list of workers was made in April 1824 at the Charenton Ironworks. There were 135 names, 84 with wives, for a total of 216 adults. Most workers were between 20 and 30 years old and came from Staffordshire.
Most births, marriages and deaths were recorded to the British Embassy and not necessarily recorded in the French records. So you might try and find who holds those records. I have heard that the Guildhall Library in London has records like that.