Good evening! This is the first time I'm approaching the stackexchange, I really hope this question doesn't sound too complex or messed up.
TL;DR: How do I create Blender armatures that work this Maya FBX's actions?
Within the project I'm currently working on, I've been asked to figure out a way to create new assets from Blender, while keeping the same structure that prior partners were using in Maya to export animated characters to Unity. The character in question has all of its actions. However, the character uses a Maya structure, and I don't know if it's supposed to look like this. 
The character itself works with the actions given. The problem arises when we need to rig a new character with this armature; as it's not symmetrical and the bones aren't connected, there may be issues and problems.
I have tried to make a new armature respecting the bone names, but the actions break with this specific armature. I have tried to rename bones within Auto-Rig Pro to match the armature, which hasn't worked either: In both cases, the bones turn to undesired directions.
Retargeting seems to give the best results, but that would imply generating new actions and assets worked and produced in Blender. In the past, we've had issues with implementing rigged FBX from Maya to our Blender pipeline, and since history's repeating itself, I would like to ask what do you think would be the best solution for this problem.
Thank you for your attention!
