We don't have synonyms for branks yet — it may be a rare, specialized, newly coined, or misspelled word. The closest thesaurus entries we have are ranks, pranks and brank.
(n) :
a punishment device, especially for scolding women, consisting of a cage to enclose the head, with a metal gag for the mouth
"1836: Plot, in his History of Staffordshire, describes the branks used at Newcastle-under-Lyme, and at Walsall, in the reign of James II. -- The Gentleman's Magazine, July 1836 p.98"
(n) :
a punishment device, especially for scolding women, consisting of a cage to enclose the head, with a metal gag for the mouth
"1836: Plot, in his History of Staffordshire, describes the branks used at Newcastle-under-Lyme, and at Walsall, in the reign of James II. -- The Gentleman's Magazine, July 1836 p.98"