Okay. I'm patched. 

It turns out that the web.config was missing a chunk of code "add name="SiteSqlServer" connectionString="Server=..." that sets up a connection string for SQL Server 2005 Express. And attempting to set the hosting space to "Full Trust" would cause a site crash because of this missing code fragment. Although I'm not sure how the heck the site worked without the code when running in Medium Trust.

It just took a bunch of wading through and comparing the problematic web.config file and the web.config file of one of the sites I manage that was already running in Full Trust and had accepted the Hotfix. I just copied and pasted the section from the working web.config file and modified the server info to match the current hosting space credentials.

And thanks to K Prouty for helping me get focused ;-)