I am inclined to think JA's comment about the unsuitableness of the connection was due to the differences in their personalities, and in Miss Churchill's wants, rather than simply being due to a difference in social status. Another woman may have been happy to leave the grandeur of Enscombe, and live happily in a small way with the man she loved (a couple of JA's heroines in other novels spring to mind), but Miss Churchill ... had resolution enough to pursue her own will in spite of her brother, but not enough to refrain from ... missing the luxuries of her former home. (chpt 2).

As for giving up his son, Mr Weston probably could do it because he felt it was for the best for his son in the long run, the Churchills could give Frank a better education than Mr Weston could afford. Indeed, we don't know how tight Mr Weston's situation was, perhaps Frank's health and even nutrition may have suffered if he had stayed with his father. We know that Mr Weston divided his time between earning money in London and his leisure days in Highbury, perhaps Mr Weston was concerned for his son's health being shifted back and forth between homes, and had no servants he trusted well enough to leave his son with. We are not told exactly Mr Weston's motives, but a great many parents throughout human history have given their children up in the hope of a better life for them.