you don't have to source one manually — pullrun kernel install fetches a kata containers kernel into ~/.pullrun/kernels/vmlinux-<version>. One-time setup, then the daemon auto-discovers it.
If you run an ubuntu image with a kata kernel, do you have difficulty installing some packages? For example if you need linux-headers-$(uname -r) or linux-modules-extra-$(uname -r), it won't exist in canonical apt repos?