Hi @Andy ,
Not only the health mailbox, but also you need to move the auditlog mailbox to another database before removing the old database.
Run below command in EMS to migrate it.
Get-Mailbox -Database databasename -AuditLog | New-MoveRequest –TargetDatabase DBname
Here's a similar case for you reference.
https://social.technet.microsoft.com/Forums/office/en-US/631ba4c4-fc03-476e-86d6-20ff1121aca5/deleting-default-mailbox-database-exchange-2016?forum=Exch2016Adm
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