Is there anything like completion for ash within Alpine containers, or do you have to install Bash?
To be fair. Bash does not have a large footprint.
As an alternative, as mentioned already in response to another comment is oksh (https://github.com/ibara/oksh), a portable version of OpenBSD's pdksh. It's in the same Bourne shell lineage as bash but is lighter and the completion is simpler: you have to customise it if you want it to work for you, but it's very easy and quick to do.