You sure? Axiom Verge and Shovel Knight were released long before Steam get DS4 support and they supposed to be working with dualshock out of the box, no?
If you want
Steam Controller functionality, you need to have it enabled in Steam.
If you don't enable it, then it's up to the game to handle controller support - and it's clear that most games with their own DS4 implementation are not compatible with the new DualShock 4s.
Since very few games have DS4 support at all - fewer still with support for the new DS4s - and the majority have XInput support, you should enable the Steam Controller functionality and get used to seeing Xbox prompts.
giving this a try, configurations took some steps with configuring the DPad. Still steam big picture mode with PS button prompts was feeling pretty good. Then I open god eater resurrection which auto recognizes a DS4 and shows DS4 prompts is now showing xbox prompts.
I'm guessing that steam is feeding the controller through as Xinput?
When you enable Steam Controller support, the games no longer see the DualShock 4.
The controller talks to Steam, and Steam's virtual device talks to the game.
The game only sees the virtual controller that Steam is presenting to them - which can be a combination of XInput, keyboard, and mouse inputs.
XInput means that you get Xbox button prompts.
There's no way around this unless the game has an option to switch the prompts manually.
What games should be doing, is
not implementing DualShock 4 support at all, but implementing
Steam Controller API support instead.
With the Steam Controller API, games automatically display the correct button prompts, just as they are mapped to your controller.
EDIT: Hmmm, then if I change the game preference and uncheck "Use Steam Configurations for PS4 Controllers" the game in settings once again recognizes "Wireless Controller" for the DS4 with the correct prompts, but no actions from the controller are registered. I'm guessing Steam is still eating the inputs
Did you disconnect the controller and wireless receiver like it tells you?
I just tried this and it worked perfectly, with Shovel Knight seeing it as a DualShock 4 and giving me PlayStation button prompts instead of Xbox prompts.
Of course you also lose all Steam Controller functionality,
since you disabled it.
And if you're using a new DualShock 4 it seems that support
in those games is broken, so it won't work properly - that's nothing to do with Steam.