He daubbed some lacquer on a wooden board and spread it with a pallet.
He rubbed some lacquer on a wooden board and spread it out with a metal spatula.
この場合、daub を使って言えますよ!
ーHe daubbed some lacquer on a wooden board and spread it with a pallet.
「彼は木の板に漆を塗ってへらで伸ばした」
to rub A on B で「AをBに塗る」とも言えます。
ーHe rubbed some lacquer on a wooden board and spread it out with a metal spatula.
「彼は木の板に漆を塗り、へらでそれを伸ばした」
ご参考まで!