With the help of Monday support, I got a functional solution for most of what I needed. The quick version is:
Connecting the Boards & Specific Items
On Board B, I set up a 2-way connection between Private Board A and Board B (I want this one to be view only for the client).
Then I set up an automation on Private Board A so that when I change status on Private Board A to X, it creates an item on Board B. This way, when I select the client on Private Board A, the project copies over to Board B. This connects the specific item from Private Board A to the item created on Board B. The part I was getting wrong was not setting up the connection first, so the specific items weren’t connecting automatically.
Updating the Statuses automatically
I set up mirror columns on Board B to reflect the statuses on Private Board A. When the status on Private Board A changes, the mirror column does as well, but, the client can’t see these because they don’t have permissions on Private Board A. So, I set up automations on Board B so that when the mirror column changes status, it changes the relevant status columns on Board B (the ones the client can see). The automation is something like: When a status changes to something, change another status to something.
I had around 7 columns, each with around 5 statuses, so this ended up being a lot of automations, but it works for statuses.
Check out Automate: Add new item in #1 GROUP A, automatically added to connected board #2 ITEM A - FAQs - monday Community for more info.
What doesn’t work
There isn’t currently a way to change dates on Board B using the mirror column that reflects the date on Private Board A. That means when we change a project due date on Private Board A, we also need to manually change it on Board B. Thankfully those don’t change much.
Hope this helps!