Hi @lindosl - I understand the issue you are facing.
When you are using a connected board column to my knowledge there is no way to pre-filter the listing of records that is returned from the connected board. What you can do is type in the name of the customer in the top bar to quickly find the record.
Outside of that organizing your customer board into different groups can make it easier to track down, as ell as ensuring the item name column in your customer board has the appropriate information in order to have a successful search.
One question I have, is it a 1:1 relationship between the parent board and the customer board? If that is the case you could setup an automation to connect these columns automatically on creation or via status change.