In this circumstance, "la" is an indeterminate pronoun and it implicitly refers to something negative.

I would argue that those two sentences have a slightly different meaning.

• finirla = "Troncare, far cessare, smettere", "a proposito di liti, contrasti, questioni noiose e sim." (Treccani, definition e.). You may specify di + <infinito> but it can also stand alone. This sounds like actively putting an end to a negative action to me.
• finire di + <infinito> = "Giungere al termine di" (definition d.). This sounds more neutral to me, as if it's reaching the natural end of an action.

Also, both sentences are rather odd: of course someone would stop asking for something once they receive it; so the conditional clause, as it stands, is meaningless.

In this circumstance, "la" is an indeterminate pronoun and it implicitly refers to something negative.

Both those sentences are grammatically correct, but I would argue that they have slightly different meanings.

• finirla = "Troncare, far cessare, smettere", "a proposito di liti, contrasti, questioni noiose e sim." (Treccani, definition e.). You may specify di + <infinito> but it can also stand alone. This sounds like actively putting an end to a negative action to me.
• finire di + <infinito> = "Giungere al termine di" (definition d.). This sounds more neutral to me, as if it's reaching the natural end of an action.

A personal note - both sentences are rather odd: of course someone would stop asking for something once they receive it; so the conditional clause, as it stands, is meaningless. It would make more sense if it were Ti comprerò una bicicletta nuova purché tu la finisca di importunarmi.