No the plugin resets the entire database as if you had just installed WordPress leaving you with just your Hello World! post and Sample Page page.
That's why I was asking if you had two versions of the site on the same server to test with, because one of them you are going to lose everything, but it will at least tell us whether the problem is that something has gone wrong, e.g. database corrupted, on your site, or whether there is a problem with server which means it wouldn't matter whether we started from scratch or not we would still have the same issue.
I don't recall you having a lot of content yet on the site.
You could export the Toolset data for your custom posts types etc etc and import them into the fresh install. But the risk is that you might export the problem, too.
Is it too much for you to manually set up again? Like, you only have one property post as far as I can see.
If you are willing to do that, be sure to re-test the issue periodically so that if it recurs you may know what triggered it.
Sorry, I have been juggling several very involved tickets and dividing my time between them.
So, to recap, you have the original site installation, which shows the problem, and a fresh WordPress install on the same server which doesn't show the problem, correct?
But on the second site there is little to no content so we can't rule out that there is still a problem with the server configuration.
Aside from getting your hosting company to further increase the memory settings and max input vars I'm not sure what alternative there is that doesn't involve moving to a different server (you could ask your host to move your site to a different physical server) or re-building the site to some extent.
Perhaps on your second site you could reproduce your Toolset settings (i.e. your custom post types, custom fields etc. etc.) and when you have done that export your content from the first site (use Tools > Export) and import it into the second site.
That should port over the lion's share of your site without simply duplicating the database. Before spending time tinkering to fix anything missing you could re-test the issue.
I was able to figure this one out, finally! I noticed that after a few days, the post field groups would show in the list. That lead me to believe there was a caching issue. Then I checked, and found that I had Memcached enabled on the server. I flushed the cache and all of my testing Post Field groups appeared!
Thank you so much for all your time spent on this with me!