Thank you for your logs and follow up.
For anyone else on WPE or another host that has lower max_execution_time settings and/or doesn't allow you to modify them... here was the conclusion on this issue:
The Cloner was really designed to quickly copy fully configured but empty (relatively no content) site templates. It wasn't really built as a full-blown copy / back-up / migration tool. It still works like you are wanting when you have full control of your hosting. For scenarios like yours we have a complex batching add-on in the works that will clone in chunks to avoid issues like inflexible max_execution_time settings. That's a ways off though.
As a first step the Cloner creates the new site and then copies all the data to it. That's why you can see the new site show up - it's there - but it's only getting partially cloned and will be broken.
In the meantime, the next best bet might be to create / configure / set up a site template (theme, menus, pages, images) but no content for cloning. Then you can copy / migrate your content separately using tools like WordPress built-in export / import xml.
That might be the best you can achieve with the Cloner today.
You could also try cleaning up your DB (deleting all transients, removing / deleting any plugins and their tables that aren't being used, removing all deleted posts / products etc.). The DB Optimize plugins out there work pretty well. Make sure you use WPE backup points :) However, if your DB is truly huge, and you have lots of product images / etc., it's still probably going to come down to just being too big.
Sorry! (also for anyone else this might affect).