There are two parts of every PII storing system. The actual PII store which is super small, "mutable" with your terminology, locked down so nobody can access it without raising an alarm and usually not accessed at all except for some very limited use cases, including GDPR ones. The rest of the store just uses references to the entities sitting in the GDPR store, like a numeric id (foreign key in SQL terminology). This way any data store, SQL, datalake, etc. can be easily GDPR compliant without needing to delete data in the large data stores and this also increases security because in case of a security breach to the data stores the GDPR data cannot be accessed.
If you tie a user to a uuid separately from where you are logging the transactions, you can nullify the existing UUID link to the given user and be in full compliance with GDPR.