Negative results, in case others are looking, also.
I couldn't find indications of LNAB for either wife of Walter Cook(e). As mentioned below (Will of William Brenton), William Brenton (of Rhode Island) had a sister Catharine Cook, but no connection to Walter Cook(e)..
- child Cooke mentioned above, born in Weymouth to Walter Cooke and Experience ---.
- Vital Records from the NEHGS Register v128:154, April 1974
Notes and Corrections section. "Catherine, Wife of Walter Cooke of Mendon," submitted by Jane F Fiske. She has "... mixing-up of facts led to the belief that she [wife Catherine] was a second wife, and Experience [was] shown to belong to someone else," referencing Chamberlain's History of Weymouth, and Snow-Estes Ancestry. I couldn't determine why Ms Fiske came to that conclusion.
- History of Weymouth, Massachusetts, in four volumes. Vol. 3. Genealogy of Weymouth Families by George Walter Chamberlain, M.S. Weymouth Historical Society. Howard H Joy, President. Under the direction of the Town. 1923. https://babel.hathitrust.org/cgi/pt?id=njp.32101067881407;view=1up;seq=5. p 174. Walter Cooke - "His first wife was Experience ---, and his second wife was Catherine ---, who died at Mendon, 2 Jan 1695-96. He died there Jan. 1695-96." No indication of LNAB for either wife.
- The Snow-Estes Ancestry, Nora E Snow, Hillburn, New York, 1939. www.archive.org. p 102-3. Mentions wife Catharine ---. Does not mention other spouse(s).
- The genealogical dictionary of Rhode Island : comprising three generations of settlers who came before 1690 : with many families carried to thefourth generation. by Austin, John Osborne, 1849-1918. www.archive.org. Under William Brenton: "1673, Feb 9 Will – proved 1674, Nov 13, at Newport where he died" - bequests “… To sister Catharine Cook’s children, [Pounds]20.” No indication that that Catharine (Brenton) Cook was wife of Walter, or was specifically located in Weymouth (or anywhere).
EDIT: Torrey's Marriages has "COOKE, Walter ... & ?1/wf Catherine [ALDRICH] (-1696) (doubtful; no evidence)." I couldn't access all his references, but found no ALDRICH. Among references: "Hill #1 53." which is Bartlett, Joseph Gardner, John Hill of Dorchester, Mass., 1633, and Five Generations of His Descendants, Also Ancestral Lines of Some Families Intermarried with His Descendants and Colonial and Revolutionary Records. www.archive.org. Couldn't determine what "53" referred to (total 22 pgs) and no individual numbered as 53.