Need help with Lookup field for Quotes

Every account has its own Distributor/Cusftomer ID text field. I have tried to make a lookup field in my Invoice layout where I chose from a list of Accounts in order to populate the Quote's Distributor/Cusftomer ID field, but it has not worked.

Is there a way to either:

To select an Account for an invoice such that a seperate field automatically populates that account's Distributor/Customer ID, as defined in the Distributor/Customer ID for that account's file.

Can I have a lookup field where i can select an account but rather than having that account's name popup, its distributor/customer ID does?