EDIT: I was going to show action sets by double tapping and holding the touch menu, but that's actually significantly more involved, so I changed the above to make things easier. Instead, we'll make it so tapping the touchpad on the DS4 take a screenshot, and physically clicking and holding down the touchpad actives the mouse.
So select the touchpad input from the steam controller config menu, and set it to activate when I touch the touchpad with my finger. Here I have it so when you touch the touchpad, it takes a screenshot through steam. Note that this works when I actually touch the touchpad, but right above that option is the ability to map clicking the touchpad to a button press.
Instead of mapping touching the touchpad to a button press, we are going to map clicking the touchpad into a
Mode shift. Mode shifting means that, when you press and hold any input on the controller, it shifts the entire controller into a second mapping. This lets you map tons and tons of mappings to a controller. Like, say a game uses normal xbox controls throughout, except the main menu needs a mouse and keyboard. You can have a mode shifting option just for menus that has mouse and keyboard controls.
To get the mode shifting options for the touchpad, go right to mode shifting:
then press enter. You'll see it ask for a type of input. This is what the touchpad will be once it mode shifts. We want it to be a mouse:
Now we need a button to use to shift this touchpad into mouse mode. Let's set clicking the touchpad to mode shifting:
The steam controller has left and right pads, so we're changing the right pad. But on the DS4, you'd only have a single touchpad click option.
So now, we can move the mouse around by clicking and holding down the touchpad, which is a separate button press than just tapping the touchpad, which takes a picture. But we'll still probably need an input to act as clicking the mouse to, for example, press a button on a menu or drag a window or whatever you need the mouse for. the L and R triggers are good for this, but we only want them to be mouse clicks when we are mode shifted. For that, we will set mode shifting options for each of those.
Go to your triggers:
in this screenshot, I already have the right trigger mapped to left mouse button. But that's not what we want, so really quickly, I've remapped it so pressing the right trigger presses the W key, just as an example:
That's now the default action of pulling that trigger, it activates the W key. Now, on the right hand side, you can see we can mode shift this input too:
Under this shifting mode, we'll make it so when we press the trigger, it sends the left mouse click:
Now this is where it all gets pretty smooth. We set the mode shift button to change the behavior of the left trigger, to the same mode shift button we used to make the touchpad act like a mouse, i.e. touchpad click:
So now right pad click changes TWO inputs -- it makes the touchpad act like a mouse when the touchpad is clicked, and it makes the trigger act like a mouse click when the touchpad is clicked. Otherwise, when the touchpad is NOT clicked, touching the touchpad acts like taking a screenshot, and pulling the trigger sends the W key. As soon as you start clicking down the touchpad button, it'll work like a mouse, and when you stop clicking the touchpad button, it'll work like a normal gamepad.
Mode shifting is like a really quick configuration option to do this stuff. More powerful are action sets, which are like mutliple modes you can shift into that can change the entire controller all at once, but those are more involved to setup. But with action sets, you have way more control, and can map double tapping and things like that. You can map any button or input, or combination of inputs on any controller to switch to action sets.
The steam controller can also do things like make pop up menus, rendered by steam, appear on screen with multiple options. These pop up menus can have menu options that, when selected, act like virtual buttons that can also be mapped like any other button on any other controller. So, as an example, you could make it so clicking the touchpad made a menu like this pop up on screen:
The little circular menu on the left side with the colorful buttons is being added by steam. Each item in that radial menu is a virtual button, and you can map those buttons just like a real button under steam. So one button could be take a screenshot, another could be maximize window, etc.
But that's also a way more advanced. For what you want, this should be an easy way to achieve it all through simple mode shifting.