Monitor Connected Through Usb-C Dock Not Detected

I have an HP Pavilion 14-dv0000sl, which has a USB-C w/ SuperSpeed and DP1.4 output (confirmed through the manual, and the USB-C port itself having the SuperSpeed and DP/thunder icon). I have bought a USB-C dock with two HDMI outputs so I can connect two monitors to the laptop. I have tested the dock on another laptop I own (Legion Y540, which also has a USB-C w/ DP1.2 output support) and the dock works on Windows 11 correctly.

Here on Ubuntu 20.04, I cannot manage to get it working for video: if I plug it with a USB wireless dongle of my mouse and the HDMI of a monitor, the mouse works while the monitor doesn't (also the laptop produces some weird electric sound when the dock is plugged in, which is troubling, but didn't do so on the W11 laptop). I have tried:

• Installing NVIDIA drivers (which admittedly didn't manage to, ended up black screening and having to remove them from root recovery)
• Blocking nouveau so the laptop would only use Intel graphics (didn't change anything)
• Waking the DP ports listed as disconnected with xrandr -d :0 --output DP-1 --auto

Nothing worked so far. Here are some additional informations:

 ~ ❯ sudo inxi -F         
System:    Host: croissant Kernel: 5.13.0-39-generic x86_64 bits: 64 Desktop: Gnome 3.36.9 
           Distro: Ubuntu 20.04.4 LTS (Focal Fossa) 
Machine:   Type: Laptop System: HP product: HP Pavilion Laptop 14-dv0xxx v: 
Graphics:  Device-1: Intel driver: i915 v: kernel 
           Device-2: NVIDIA driver: N/A 
           Display: server: X.Org 1.20.13 driver: modesetting,nouveau unloaded: fbdev,vesa 
           resolution: 1920x1080~60Hz 
           OpenGL: renderer: Mesa Intel Xe Graphics (TGL GT2) v: 4.6 Mesa 21.2.6

~ ❯ xrandr  
eDP-1 connected primary 1920x1080+0+0 (normal left inverted right x axis y axis) 309mm x 174mm
HDMI-1 disconnected (normal left inverted right x axis y axis)
DP-1 disconnected (normal left inverted right x axis y axis)
DP-2 disconnected (normal left inverted right x axis y axis)
DP-3 disconnected (normal left inverted right x axis y axis)
DP-4 disconnected (normal left inverted right x axis y axis)

 ~ ❯  sudo lshw -c video
  *-display                 
       description: VGA compatible controller
       product: Intel Corporation
       vendor: Intel Corporation
       physical id: 2
       bus info: pci@0000:00:02.0
       version: 01
       width: 64 bits
       clock: 33MHz
       capabilities: pciexpress msi pm vga_controller bus_master cap_list
       configuration: driver=i915 latency=0
       resources: irq:149 memory:56000000-56ffffff memory:60000000-6fffffff ioport:5000(size=64) memory:c0000-dffff memory:59000000-5fffffff
  *-display UNCLAIMED
       description: 3D controller
       product: NVIDIA Corporation
       vendor: NVIDIA Corporation
       physical id: 0
       bus info: pci@0000:01:00.0
       version: a1
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress bus_master cap_list
       configuration: latency=0
       resources: memory:57000000-57ffffff memory:40000000-4fffffff memory:50000000-51ffffff ioport:4000(size=128)

~ ❯ lspci -k | grep -EA3 'VGA|3D|Display'
0000:00:02.0 VGA compatible controller: Intel Corporation Device 9a49 (rev 01)
    DeviceName: Intel Tigerlake UHD Graphics
    Subsystem: Hewlett-Packard Company Device 884e
    Kernel driver in use: i915
--
0000:01:00.0 3D controller: NVIDIA Corporation Device 1f97 (rev a1)
    Subsystem: Hewlett-Packard Company Device 884e
    Kernel modules: nvidiafb, nouveau
0000:02:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. RTS522A PCI Express Card Reader (rev 01)

EDIT: I've jumped through numerous hoops trying to get the NVIDIA drivers to work on my machine and see whether video through USB-C would work afterwards, but I failed. I've tried:

• Installing from Additional Driver -> Black screen after GRUB
• Installing either the normal version or the server version of the recommended driver from command line -> Black screen
• Using nvidia-prime, setting options nvidia-drm modeset=1, adding nomodeset kernel parameter, just working through an innumerable amount of questions on this issue

While searching around, although not sure about it, I found that the issue might be related to LUKS encryption of my disk with the NVIDIA drivers, such as this bug and this one. I'm giving up for the time being on being able to use the dock on Ubuntu, sadly.