Just to leave a note for someone who would come across
As I suspected, the issue was in device tree configuration. I checked DTS files for the rock 4c+ in the corresponding release tag of the mainline kernel source (6.1 in my case), and saw that it had bluetooth config present, similar to Radxa's own kernels. I could not find where does Armbian take their DTSes from.
Anyway, things I did:
Followed regular steps for building the kernel. To be safe, I went with version of the sources as close to the running kernel version as possible.
Built just device tree blobs with `make dbts` in the root of sources.
Copied over the generated file (arch/arm64/boot/dts/rockchip/rk3399-rock-4c-plus.dtb) to /boot/dtbs
Updated `/boot/armbianEnv.txt` to point `fdtfile` the blob I just put there.
Reboot!
After this I immediately got bluetooth and sound working, I do not experience any issues with BT keyboard connectivity/speed (as I used to with official legacy 4.4 kernel).
P.S.: there are though some things that got broken because of the update. Particularly, I cannot seem to be able to enable the thermal sensor ADC anymore, although, it works fine with DTB shipped by default.