We then tried to access it via its host and a "hosted-engine --console"
but ran into an
internal error: cannot find character device <null>
which I know from KVM. With other VMs I could follow the RedHat's advice
to add a console
(https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/6/html/virtualization_host_configuration_and_guest_installation_guide/app_domain_console)
but although I edited the hosted engine's profile those changes weren't
applied - and after an engine restart even deleted again. That kind of
makes sense to me, but limits my options.
with engine 4.2.5-1 you can definitively enable the serial console device editing the engine VM from the engine itself.
If you are not able to reach the engine VM for the upgrade, you can copy /var/run/ovirt-hosted-engine-ha/vm.conf somewhere, edit it to add the console device and restart the hosted-engine VM with a custom vm.conf with something like:
hosted-engine --vm-shutdown --vm-conf=/root/my_vm.conf
As an alternative, VNC console should work as well; you can set VNC password with
hosted-engine --add-console-password