Update: I may have figured the issue out. Over the course of further troubleshooting, I decided to remove and reinstall the OpenTouch web service on our test server. Once I did this and re-tested my script, I received a new error stating that the account I was using did not have rights to read Process._ColourHTML.
Even though I had tried to use the Priviledges wizard earlier to enable all rights from the top of the Modules tree down before, the Read permission on Process._ColourHTML was still disabled. Enabling it seems to have fixed everything so far, but I will update this post again if something changes during further testing. Hope this helps if someone else hits a similar problem.