Compiling UVC kernel module out of source tree against running kernel - linux-kernel

Question

I am trying to make modifications to the UVC kernel module and then insert it into my running kernel. To do this, I performed these steps:
Install linux-headers with sudo apt install linux-headers-$(uname -r)
Checkout the UVC drivers folder from the Linux kernel GitHub repository
Enter directory and try to compile with make -C /lib/modules/`uname -r`/build M=$PWD as specified by Building External Modules from the kernel.org docs.
But I get this error from Make, complaining that it can't find an include from the tree above (which I do not have checked out)
me#bionic:~/src/media/usb/uvc$ make -C /lib/modules/`uname -r`/build M=$PWD
make: Entering directory '/usr/src/linux-headers-5.4.0-131-generic'
CC [M] /home/me/src/media/usb/uvc/uvc_driver.o
/home/me/src/media/usb/uvc/uvc_driver.c:23:10: fatal error: media/v4l2-uvc.h: No such file or directory
#include <media/v4l2-uvc.h>
^~~~~~~~~~~~~~~~~~
compilation terminated.
scripts/Makefile.build:270: recipe for target '/home/me/src/media/usb/uvc/uvc_driver.o' failed
make[1]: *** [/home/me/src/media/usb/uvc/uvc_driver.o] Error 1
Makefile:1762: recipe for target '/home/me/src/media/usb/uvc' failed
make: *** [/home/me/src/media/usb/uvc] Error 2
make: Leaving directory '/usr/src/linux-headers-5.4.0-131-generic'
But shouldn't it be able to find this include from my running kernel (given by -C /lib/modules/`uname -r`/build)? How can I compile this kernel module?
The Makefile from inside the UVC folder contains:
# SPDX-License-Identifier: GPL-2.0
uvcvideo-objs := uvc_driver.o uvc_queue.o uvc_v4l2.o uvc_video.o uvc_ctrl.o \
uvc_status.o uvc_isight.o uvc_debugfs.o uvc_metadata.o
ifeq ($(CONFIG_MEDIA_CONTROLLER),y)
uvcvideo-objs += uvc_entity.o
endif
obj-$(CONFIG_USB_VIDEO_CLASS) += uvcvideo.o
and the Kconfig file contains:
config USB_VIDEO_CLASS
tristate "USB Video Class (UVC)"
depends on VIDEO_V4L2
select VIDEOBUF2_VMALLOC
---help---
Support for the USB Video Class (UVC). Currently only video
input devices, such as webcams, are supported.
For more information see: <http://linux-uvc.berlios.de/>
config USB_VIDEO_CLASS_INPUT_EVDEV
bool "UVC input events device support"
default y
depends on USB_VIDEO_CLASS
depends on USB_VIDEO_CLASS=INPUT || INPUT=y
---help---
This option makes USB Video Class devices register an input device
to report button events.
If you are in doubt, say Y.