Unfortunately, at the moment it is built on .NET Framework and WebForms, so we can't use IdentityServer4. IdentityServer3 would have been a possibility, but it's gone out of support and therefore isn't an option.
I believe you are right about the configure being called for each instance of the service.
It appears as though the issue may have been a combination of several things:
Once I fixed the database security issue, increased the timeout for the Lambda's from the default 30 seconds, upped the memory limit for the Lambda from 256 to 512, and fixed the log4net configuration, I was able to get the system to start returning some requests.
I will be continuing to work with this configuration and will write another update once I know more.