Sorry, my guess is that you may need to ask some PS4 forum or the like about this. You might also ask on Arduino forum as Pro Micro is an Arduino thing and maybe someone there has a solution.
Again I am not an expert with full PS4 internals (don't have one), I have had (and I think still do) have a PS3, but we then went to XBox...
And I know that for example the XBox would not use the Logitech wheel, you had to get one specifically fro the XBox. My guess is PS4 is similar. They probably have some additional checking and or protocols they are using that the hand brake control is not doing.