The baptism of Robert's son George Rice Cottrell, 7 Oct 1827 at St Olave, Exeter, does clearly give Robert's middle name as Carew:
http://search.findmypast.co.uk/recor...hlights=%22%22
Occupation Last Maker, abode Fore Street, and Robert's wife is Sarah, so he is the Robert Cottrell who married Sarah Rice 1 Feb 1814 at West Teignmouth, although no middle name is given on the records for that marriage on FamilySearch (I haven't found an image.) The other children's baptisms don't show Robert's middle name.