I managed to get what I need by compiling a new (hopefully) compatible set of modules and installing by hands only the needed ones. Here are the steps I followed, executed directly on the AXON as root:
v=$(uname -r)
cd /usr/src
wget https://cdn.kernel.org/pub/linux/kernel/v4.x/linux-$v.tar.xz
tar xf linux-$v.tar.xz
cd linux-$v
gzip -dc /proc/config.gz > .config
apt install libncurses5-dev bc
# Customize the kernel to your needs. In my case I enabled the
# snd-usb-audio module that, in cascade, enabled other sound stuff.
make menuconfig
# Build and install *only* the new modules.
make modules
mkdir -p /lib/modules/$v/kernel/sound/usb
cp sound/usb/*.ko /lib/modules/$v/kernel/sound/usb/
cp sound/core/snd-*.ko /lib/modules/$v/kernel/sound/core/
depmod
# Not strictly needed, but forbidding future axon-kernel upgrades
# could avoid potential hard-to-debug problems.
apt-mark hold axon-kernel
