Just an update to this. I opened a case with VMware who was quite prompt in confirming that this is an issue (though 'bug' may be a misnomer). It is due to the fact that PowerCLI (and specifically the connection manager) are intentionally designed to assume a single 'AppDomain' per PowerCLI instance. This works well most of the time since every individual Powershell console is its own AppDomain. However, ISE uses a single AppDomain for the entire application, even though it creates different runspaces for each PowerShell tab. Because of this, there are collisions between the Powershell tabs in ISE for VIServer connections.
They recommend that if one wishes to use ISE, then do so just for script development. Then either use multiple instances of ISE for different vCenters or actually test/execute the scripts in individual Powershell consoles.
VMware said they will mark this as an issue, though a low-priority one (thus not demanding a patch or updating older versions of PowerCLI), and will seek to rework this area to fix the issue in an upcoming release.
Thank you LucD for helping me confirm this issue.