Thank you linuxvangog! Your advice led me to a solution which I shall explain here to aide others who may have the issue I have had figuring this out.
I have set up Bluetooth and resolved an issue where the Wii U Pro Controller is treated as a mouse device by X.org. A special rules conf file defines overrides to ensure the input from this device is not to be confused or interpreted as a mouse. That aside, Transisitor will start up and identify that such a device exists, but it will still resort to the generic mono joystick driver, which doesn't work. There's no response from the gamepad within the game.
The SDL config that linuxvangog provided is the solution. There's an entry there for Wii U Pro Controller for Linux, and for some reason the definition was incorrect with A and B buttons swapped, and X and Y buttons swapped. I corrected the definition yielding the following line:
050000007e0500003003000001000000,Nintendo Wii U Pro Controller,platform:Linux,a:b1,b:b0,x:b2,y:b3,back:b8,start:b9,guide:b10,leftshoulder:b4,rightshoulder:b5,left stick:b11,rightstick:b12,leftx:a0,lefty:a1,rightx:a2,righty:a3,lefttrigger:b6,righttrigger:b7,dpup:b13,dpleft: b15,dpdown:b14,dpright:b16,
Al the other buttons matched up just fine. Regarding the config screen within the game: you cannot select the gamepad icon with the mouse. Simply press a button on the gamepad. The game will respond to this input and switch itself over to gamepad config mode by itself. You will immediately notice the game also replaces all button icons with the buttons from a controller (looks like the Xbox 360 gamepad pattern actually, but that coincidentally aligns with a Wii U Pro Controller without confusion.)
The game's native gamepad support changes how input is handled and assigned, so this is definitely preferred over using a mouse/keyboard mapper to the gamepad. Using the SDL definition above, the game boots identifying the controller and use the special appropriate driver. The following line should resemble what is shown in output when starting the game:
Controller 0, Nintendo Wii U Pro Controller, will use SDL_GameController support.
Final note: the way the setting must be consumed from the gamecontrollerdb.txt file. It's an environment variable consumed by SDL in general. With the aforementioned file name, use the following in the shell:
$ export SDL_GAMECONTROLLERCONFIG=`cat ./gamecontrollerdb.txt`
Obviously for a persisting effect it may be desirable to put this export statement (and perhaps substitute out the inline cat call) into the shell profile file (e.g. ~/.bashrc for Bash shell).
I'm happy to have this one worked out, and the game's native gamepad support is definitely well defined. After a quick test, with the native support, it doesn't look like an analog stick controller is necessary, and I could probably use my Gravis Gamepad Pro if I wanted to. I couldn't tell that the right stick was needed and the d-pad works just as well as the left stick. Note that this does not apply if the input configuration is to map mouse/keyboard to a gamepad where it seems like dual sticks would be essential.