Things *seem* to be going well!
The kernel sources were taken from Amazon's website at https://kindle.s3.amazonaws.com/Kindle_src_5.3.7.3_2715280002.tar.gz . When compiled and executed, it will display '2.6.35' as its version number, but it is in fact a 2.6.31 kernel. The reason for this hackery is that the glibc chroots in InkBox OS don't support 2.6.31 kernels, but do support 2.6.35.x ones. Changing the version number in the kernel Makefile seemed to do the job fine, and I haven't seen any problems in using glibc binaries and libraries, they all run fine with this new 'patch'.
This kernel has been slightly modified to support devtmpfs mounting via backporting and with some changes due to the 2.6.35 hack mentioned above, especially in the AR6003 Wi-Fi chip sources.
A new toolchain (arm-none-linux-gnueabi, GCC 4.4.1 from 2009) has been added in the toolchain/ directory and is the only toolchain I know that produces a bootable kernel.
You may compile a kernel for this device by executing the following command:

env GITDIR=/home/build/inkbox/kernel TOOLCHAINDIR=/home/build/inkbox/kernel/toolchain/gcc-4.4.1/ THREADS=8 TARGET=arm-none-linux-gnueabi scripts/build_kernel.sh kt root

Taken from @akemnade's https://github.com/akemnade/u-boot-fslc/tree/kobo-2020-10 tree.

The kernel sources have been taken from there, thanks to @akemnade: https://github.com/akemnade/linux/tree/kobo/merged-5.16
Commit: 40a69e57c8491d6a90224672e7501ca2d0022550

This was a nightmare, took my entire evening and a part of the night, but I'm glad that it works!

(We had some trouble with the pull request, so I merged the changes in one commit here)
Closes #2

Error code 6 is already used by https://github.com/Kobo-InkBox/rootfs/blob/2c45339df177aae26fcdc321484806ad5ec793b8/etc/init.d/inkbox_gui#L100-L103