I know of three ways to get your controller working in wine

1. x360ce (didn't try that one as it only maps controller input to buttons)
2. dumbxinput
-make sure to get the right files for your wineprefix (32 or 64bit)
-you have to extract xinput1_3.dll in the same folder where the games .exe file is located.
-open winecfg for your prefix and add the library as native
-try if the controller works now
-if not, try to extract xinput9_1_0.dll as well
-depending on your wine version it "might" be necessary to create copys of xinput1_3.dll, name them xinput1_1.dll ...2.dll and ...4.dll and add them to the same folder.
3. there's a wine-patch for xinput here
the good thing is this one worked for me where option 2 didn't (witcher 3)
the bad thing is you have to patch wine source and compile it yourself, bad that sounds harder than it really is ;)

I've already tried 1, it didn't work.
dumbxinput I just tried as well, didn't work either.
The wine patch I'm not sure how I'd use that, since I'm using PlayOnLinux which sits between me and wine.

And while I'm grateful for all that info, that still doesn't really tell me how to get a controller running from a clean slate, since I don't know which other wine components and libraries I can install and which will ruin it again.

Playonlinux is just a Frontend for managing prefixes, multiple versions of wine and installer scripts. I'd rather usw lutris if anything at all. But nevermind, what game are you trying to play with a controller?

Quoting: leodI've already tried 1, it didn't work.

I've finished DIRT 3: Complete Edition using DualShock 4 through x360ce and had not a single issue. With PES 2015 I had small problems while calibrating controller (triggers were recognized as triggered all the time) but after around 15 minutes of playing with buttons I managed to configure it.

What game do you try to play with your x360 controller?

I was trying Nidhogg 2, Crawl, Dead Cells and Project64.