Thanks for this! I think this is a new setting. If you go into Early Access Apps, you can enable the Core Data Access Permissions, but it's not clear what this actually enables.
Here is the URL for this and it specifically mentions Bard:
https://apps.google.com/supportwidget/articlehome?hl=en&arti...
You need to click into "Early Access Apps" from the services list, which takes you to the service's settings page.
Did that and I still don't have access... is there more to do or is there just some random wait?
EDIT: it's working now, took about 10 minutes.
I think that simply enabling "Early Access Apps" isn't enough. Once you enabled this, you need to add a checkmark to signal that you "Allow users at your organization to access Google Workspace data and services using Early Access Apps".
In my case enabling these two things was nearly instantaneous, it told me it will take a couple of minutes (located in Germany).
A sibling comment to this one contains the link to the help.
Sorry for not providing the direct link to the setting. Even though my domain name was not in the URL, there was a long number in it which might be personalized (albeit with a very low probability of this being the case).
And as another commenter predicted, searching the admin console for "Bard" turns up no settings.