Thanks, I have knew her father was "of Hartwell", presumably to differentiate the family from the Hanslope Coxes.
I do not believe the Rainbows have any connection with the Cox family.
When I looked at the Hanslope registers back in the 1960s, I noted that the father of Sarah Ann Cox, Joseph Cox, was described as a labourer OF HARTWELL. (This does not appear in the transcript).
There is a Joseph Cox / Sarah Ann Cox pairing in the 1861 census for Hartwell (RG9-927-49-1) in which her age is given as 12, and Sarah Ann reappears (as Bissell) in the 1871 census for Wolverton (RG10-1475-22-36) in which her age is given as 23. So I deduce that her real age at marriage must have been about 17-18 and not 20 as shown in the register.
I have never attempted to trace Joseph Cox's ancestry, but it may have connected with the Coxes of Hanslope at some earliier date - perhaps you have already researched that line?