Short answer here is that whomever is calling that function needs to ensure that their search_path is setup so that the type can be found somewhere in it. Your desire for obscurity means you are pretty much on the hook for figuring out the right command to do so.
I hope this is the right list for me to ask questions about psql. Please let me know if I am in the wrong place. :)
Right place
When listing the functions, I see that functionName() does exist in the database.
As the error is coming from within the function it indeed must exist and be visible to you.
The type also exists (I think):
# select exists (select 1 from pg_type where typname = 'xxx_xxx_xxxxx'); exists -------- t (1 row)
Existence and visability are two different things. It indeed exists. It is apparently not visible to the user when at the time the function is invoked - and the function doesn't explicitly say where to find it.
Note that the role that owns the 'type' is not the same user that is calling the "functionName()" from the psql prompt.
Doesn't matter. Types in PostgreSQL are not restricted since they never themselves contain any data. As long as you can find a custom type you can use it.