How can I export a shared mailbox to PST in new Outlook?

Try adding the mailbox manually as a shared folder or mailbox. Right-click your mail account in the folder list, choose Add shared folder or mailbox, and select the shared mailbox. It may then appear as a source when you export to PST.

That is different from trying to add it as a standalone account with your own credentials.

If that doesn't do it, classic Outlook is another option for a one-off export. Check how much mail is available locally before exporting. If Outlook is only keeping a limited period offline, you may need to change that setting and wait for the older mail to download.

Without admin access, Purview eDiscovery isn't something you can run yourself. If the client has someone with the right access and licensing, they could look at exporting it that way instead. That avoids relying on Outlook staying open for a local export.