Output of "Get-PSSnapIn -Registered" (service running as Local System):
Name : Microsoft.Exchange.Management.PowerShell.E2010
Name : Microsoft.Exchange.Management.Powershell.Support
Name : Quest.ActiveRoles.ADManagement
Name : SqlServerCmdletSnapin100
Name : SqlServerProviderSnapin100
Name : VMware.DeployAutomation
Name : VMware.ImageBuilder
Name : VMware.VimAutomation.Core
Name : VMware.VimAutomation.License
So it does see them. Just can't load it. Only the first snap-in produces that error (so the other Exchange snap-in does load correctly). As Brian points out, there are no 32-bit versions of Exchange 2010 software (and all servers involved are 64-bit).
However, if I run the VisualCronService.exe manually, it does load the snap-in successfully. It does this successfully whether or not I specify credentials in VisualCron, as long as the user is (at least) a local administrator for the server.
If I then run the VC Service under a newly-created user who is a local administrator for the server), I can also load the exchange plug-ins.
I don't understand why, but at least I've got a workaround now. Thank you for the assistance.