This rare kanji has no clear standalone meaning. Treat its meaning as uncertain.
The exact historical development of 桰 is uncertain. It visually contains 木 (tree) on the left and a right side combining 口 (mouth) and 氏 (clan), but the original semantic and phonetic roles are unclear.
With no clear meaning, picture a wooden (木) tablet where a clan (氏) speaks (口) — perhaps a record or tool, but its purpose is lost.
For カツ, imagine a wooden (木) cutting board used to prepare katsu (カツ) cutlets: the sound of katsu echoes the reading while the wood ties to the left component.