German does not have SVO word worder. It's V2 in declarative clauses and SOV in dependent clauses. V2 includes SVO, but it's not limited to that.
You may use OVS word order in declarative clauses whenever you wish. It does not depend on the case being distinguishable. If there's ambiguity, its up to you to add information to make things clear — or leave them unclear by purpose. It doesn't have to be an accusative object either.
Markus antwortete ich: „Dann eben nicht!“
Here, Markus is an unmarked leading dative object. It's clear from context (ich is nominative) who's the subject.
Der Lüge verdächtige ich dich!
Thats a leading genitive object, verb second, subject, accusative object.
Ein Hund ist er!
That's a leading Prädikativ ("nominative object"), the copula sein, and the subject.
About the nicht, the closest to English you can get is as follows.
No, she does not want a different one.
Nein, sie will einen anderen nicht.
Nein, einen anderen will sie nicht.
These are both possible but the first one sounds unnatural if sie isn't emphasized in speech. Remember the rule about word ordering and emphasis? The first item gets the most emphasis, the last second most. The last is nicht. But the thing you want to express isn't sie–nicht but einen anderen–nicht. That's why einen anderen should be emphasized and therefore, has to lead the clause.
Nein, einen anderen will nicht sie.
This is also possible but it negates sie instead of the whole clause. People would ask "Then, who wants a different one, if not she?"
Nicht negates the following item. If it's at the end of a clause, it negates the whole clause.
If you want to put it plain, without special emphasis, use the pronoun kein. It's especially made for that purpose.
Nein, sie will keinen anderen.
Nein, keinen anderen will sie.
Here, the emphasis is split almost equally sie—keinen anderen, with only a slight advantage on sie resp. keinen anderen.