Though I do agree that the middle initial is "F" and the last name of a mother this was a very common thing back then for children, but I was unable to personally locate the whole middle name spelling.
Your explanation is a logical one for the whole middle name, I was only able to locate that it was "F" along with the following Basic Information.
Name: Robert F. Leverett
Birth: 16 Mar 1805 - South Carolina, USA
Marriage Abt. 1827 - South Carolina, USA
Death: 8 Mar 1893 - Texas, USA
Father: Thomas Leverett
Mother: Temperance Floyd
Spouse: Martha Pitts