Continuing on from the previous post with the changes suggested by the hosting provider:
The default wp-cron.php job was disabled in favour of a server-level cron job which runs twice per hour.
Gzip compression was enabled on the server.
Metrics/resource usage were checked on the server side; to be monitored regularly.
In about a week, the hosting provider will look over the situation again.
This completes the recommended updates to address the problem.