"A dollar" is the complement (object) of the preposition "for".
You are correct: you can't move it around; its position is fixed immediately following the preposition "for", and it's not an indirect object, though "his neighbour's lawn" is direct object of "mowed". The transitive verb "mowed" is called 'monotransitive', because it can only take one object, a direct object.
The function of "a dollar" is 'complement' of the preposition "for". Preposition complements can take several forms, one of which is a noun. A noun as a complement of a preposition is said to be the 'object' of the preposition. But be careful, the object of a prep is NOT the same as a direct object of a verb. Compare:
(1) I was visiting Ed. ("Ed" is direct object of the verb "visiting")
(2) I was talking to Ed. ("Ed" is object of preposition "to")
It's important to grasp the distinction between those two kinds of 'object'.
Now, here are two golden rules to remember:
(1) A noun which follows a preposition is always the object of the preposition and never an object of the verb.
(2) When a verb is monotransitive, the object is always the direct object, never the indirect object.