There's a Power-Up called External Share that will let you do this. Install it on your Trello board, then create a share link with a filter with the lists you want to show your customer.
Note that it is a paid Power-Up that will cost $3 per month - a bonus is that you get access to the other 15 Power-Ups listed here as well for just that one single price.
Thank you Adam, for your suggestion! But I want to share the entire board with the customer except for one list. Trello sent me a message that this is not possible (yet). They will consider implementing it. In the meanwhile, I will make a separate board with this private info just for our employees.