The Green Jedi are an official Order, they just stay in the Corellian System and are affiliated with the Corellian government because of their history. The Altisian Jedi were *founded* by a Coruscanti Jedi Master, who felt that the Order's strictures were too stringent. He's trained hundreds of students in the last few in story decades and not one of them have Fallen. The Outer Rim has probably seen more of them than they have of Coruscanti, despite there being orders of magnitude more of the latter. Finally, the Coruscant Order's connection to the government is a bad thing. It clouds their connection to the Force, and means that they bow to political expediency more than they should. That's also canon, as you can see with their treatment of Ahsoka's trial.
Just because Anakin is being trained by a Dark Jedi doesn't mean he's going to become a Dark Jedi. Even if he does, it doesn't mean he's going to immediately become Cackles von BabyEater. The Coruscant Order have become so calcified that they don't think returning from the Dark is possible, but it explicitly is. If we the players want Anakin not to Fall, we'll have the ability to influence and make that choice.
As to whether the non-Coruscanti Orders would accept a dark sider? Probably not without reservation no. They'd try and redeem him though if he wasn't a sociopathic monster. The main reason that Dark Siders get the snicker snack treatment is because they're also HOMICIDAL SOCIOPATHS. If Anakin honestly wants to come back into Balance and turn from the Dark? Most Jedi would try and help him, even the Coruscanti. Now, the Coruscanti wouldn't necessarily train him afterwards, but then again, maybe they would. Jedi Shadows deal with crazy stuff. Quinlan Vos, Aayla Secura, and Master Tholme all had issues with the Dark Side and were allowed back into the Order.