No, I do not believe that the films content supports this case, with Cutter only becoming empathetic towards the Bordens once he realised that 'Angier' was prepared to allow the execution to go ahead despite the self evidence of innocence, a final vindictive victory that included the satisfaction derived from him taking custody of Borden's daughter.
It appears that Cutter knew Angier and Julia prior to him hiring Borden as a second shill and that he sided with Angier due to Julia's death. He makes it clear that he felt Borden deserved the injury he sustained during the bullet catch and upon seeing Borden's growing success following the premier of 'The Transported Man' he worked as hard as possible to help Angier's success, correctly guessing Borden's method from the get go.
Although he gradually became more and more cynical of Angier's obsession, including his refusal to accompany him to Colorado, he was still prepared to assist in 'Fallon's' abduction and Angier's final run of shows leading up to the murder. Upon the reveal of Angier's identity he realises that as well as being misled for the totality of his partnership, he had been manipulated into becoming party to the framing of Borden and in an effort to make at least partial amends, he provided 'Fallon' with the location of Lord Caldlow and the prestige 'materials' he had used to enact his revenge, with the obvious implication of him learning the truth about Borden and Fallon's real relationship.
While not a primary source, the synopsis provided by Wikipedia also seems to take this position stating
When Cutter realises that Angier is still alive, he is disgusted that Angier allowed Borden to be sentenced...
I apologise for what may seem like a purely subjective answer but many discussions such as this one can be somewhat objectified by the films content and your question raises a genuinely valid query, one that I only perhaps have more insight into due to watching the film countless times since its 2006 release.