If Tesla had nothing to do with Bale's trick, the fact that he was able to coincidentally build that machine makes no sense.
He claims that he was out of money, so builds it anyway even though he never had before. However seems like a huge coincidence that he could.
It works.
Borden sent Angier there as a wild Goose chase but he didn't anticipate that Tesla would legitimately be able to help him. Telsas name helped legitimize sending him, but Borden didn't realize just how committed and brilliant Tesla would be at the hands of unlimited funding from "Lord Caldwell" for this project.
It's not merely a coincidence... Bale inadvertently set the whole thing in motion. If Angier doesn't go there, Tesla never even attempts it.