turban, kerchief, veil, wrap
帕 centers on a piece of cloth used to cover or wrap the head: a turban, kerchief, or veil. The wrap sense extends naturally to any cloth wrapping.
帕 combines 巾 (cloth) with 白, which likely contributes a phonetic element. The exact historical development is uncertain, but the cloth radical clearly points to fabric items.
The left side 巾 is a cloth, and the right side 白 is white. Imagine a white cloth wrapped around the head as a turban or veil.
For バツ, picture a turban wrapped so tightly it makes a 'bats' sound when you pat it: bats -> バツ, and the cloth turban is secure.