You are correct in that it is not doable nor suggested to migrate H2 to PostgreSQL. I suggest just pointing to your new database by simply editing
$SONARQUBE_HOME/conf/sonar.properties to point to the new DB. SonarSource only offers a DB copy tool for customers with Support, see here.
There are possible data integrity issues that you will have to verify yourself but there is nothing stopping you from attempting it with 3rd party tools like Full Convert. I found Metabase, which also gives instructions for your conversion (this is only a suggestion and not a recommendation): https://www.metabase.com/docs/latest/operations-guide/migrating-from-h2.html
In fact, with SonarQube v8.4 (latest as of this posting), this is the warning that is shown at the bottom of page when you navigate through SonarQube:
By the way, SonarQube v7.3 is old and no longer supported, so I would at least upgrade to the v7.9 LTS version first, then upgrade again, if you so wish, to the latest v8.4 version. Follow the upgrade guide if you’re interested. This could also simplify your backup/restore process.