If anyone encounters a similar issue, we’ve just found a way around it that worked for us.
We installed QSM but left it deactivated, then used WP Rollback to downgrade to a much earlier version (I think we started at v5.0). This installed and activated successfully, and we were then able to upgrade to the latest version. We did it gradually using WP Rollback, but you may be able to go straight to the latest version in one jump.
