Hi Sjoerd,
Try adding the assignment to the "More Assignments" list.
ie set up the lookup as-if you were gonna store the unique code part. So value = Id, Desc = Name and so on.
Then in the more assignments part set the Inv:Customer = Cus:Name.
(I'm intrigued that you're storing a non-unique part of a lookup record, but that's not really material to the answer.)
cheers
Bruce