Individuals H and J are 2nd cousins (supported by a strong documentation trail) who share two ancestors: John Charles Wright (circa 1830-1904) and Caroline Ellen Brown (circa 1827 – 1871). They share 223cm of DNA which is comfortably within the range of 2nd cousins (DNAPainter).
H and J have a common match (on Ancestry) D who descends from Henry Dillistone Brown born 15 May 1852 in Sawbridgeworth, Hertfordshire, illegitimate son of Caroline Ellen Augusta Brown (also supported by a strong documentation trail). They share 24.3cm which is (just) feasible for half-second cousins (which the documentation suggests.) (also DNAPainter)
I have double-checked the documentation trails and am happy that they are robust (including determining that the mother of Caroline Ellen Augusta Brown was resident in April 1861 at the address at which Caroline Ellen Brown gave birth to my ancestor in May 1861, and that Ellen the mother of Henry Dillistone Brown signed his Royal Navy enlistment papers as Ellen Wright.) Although I'm comfortable that the two Carolines are the same person, would these DNA results unsupported by the smoking gun of the mother's residence be sufficient to conclude that Caroline Ellen Augusta Brown and Caroline Ellen Brown are the same person or would I need to look for further DNA evidence? If so, how would I go about it?