The bill must be paid by whoever purchased the lumber.

A. do-direct object

B. op-object of preposition

C. sc-subject complement

D. s-subject

The whole clause acts as a noun. So, as a noun that follows the preposition, it is the object of the preposition.

A direct object after your verb would be more like: The girl must be paid a hundred dollars. Paid what? a hundred dollars. The preposition in your sentence makes the difference.

So, in this case, the prepositional phrase (that begins with "by", and ends with the object of the preposition which is a noun clause) describes the main verb "paid", and is therefore an adverbial phrase.

The prep phrase does describe the verb, but the noun clause within the phrase, functions as the object to the preposition.