While it is difficult, if not impossible, to prove a negative, the existing records do not support Fairleanah Eleanor Harrison (Wheeler) as a daughter of Benjamin Harrison V (of Berkeley) and Elizabeth Bassett Harrison. Seven children of this union survived into adulthood and are mentioned in the Will of The Signer:
1. Benjamin Harrison VI
2. Carter Bassett Harrison
3. William Henry Harrison
4. Elizabeth Harrison Rickman (later married Edmonson)
5. Lucy Harrison Randolph
6. Sarah Harrison Minge
7. Ann Harrison Coupland
I do not believe Ann Harrison mentioned at 7 above could be the Fairleanah Eleanor Harrison Wheeler aka Anna mentioned by the second poster. Ann Harrison Coupland died a Coupland and her will (WB 2, page 503) indicates her residence as Charles City Co.
It may be worthwhile to research Benjamin Harrison of Prince George County, Virginia (Benjamin of Brandon). Benjamin of Berkeley and Benjamin of Brandon were cousins of some distance (both decedents of Benjamin of Wakefield) and contemporaries. Benjamin of Brandon’s second and third wife were both named Elizabeth. Benjamin of Brandon had at least 6 children, two sons: William and George and four daughters whose names are not readily available to me. Perhaps this Benjamin/Elizabeth Harrison could be the connection you seek.
In this context “Berkeley” and “Brandon” refer to land holdings along the James River and not to Berkeley County, West Virginia.