"But" as "Without" seems to mean Outside rather than the usage of "Not Having". In use in Scotland as "But and Ben" - a small cottage consisting of two rooms, the "Ben" - inside/living room (within) and the "But" - kitchen/outside room (without).
I don't see any connection between that particular usage and "without a hat" which would justify "baht" as deriving from "but" in the within/without usage.
B'aht is a shortening of (with)'(out) which in modern "town" Yorkshire would be more likely pronounced "wi'aht".
Quack!
GtD.