I wasn't able to find this layout, maybe because our controllers are differerent. Right now I'm trying this:
Apply the official warframe template for PS4 (I first tried xbox, didn't seem to do anything in-game)
In every configuration. replace the right stick with Joystick Mouse (or flickstick or whatever your preference)
In every configuration, replace L2 and R2 with Right and Left mouse buttons.
Camera has been behaving and moving at a predictable sensitivity, I can access all the gamepad buttons in all the menus as long as I remember them. I've mostly been running around in the orbiter trying this.
As soon as any mouse movement is detected by the game, gamepad look seems to be completely disabled, so you have to replace it wholesale. Not a particularly big issue.
You will sometimes get UI flickering, but that's just the gyro life.