As it turns out, I did actually already find information about how to do this but I had misinterpreted something so it didn't work for me before.
There is a method for accessing a datastore via PowerCLI as a regular drive so you can use any of the methods that you would usually use (Remove-Item, Rename-Item, etc.). This is done using a PSDrive,
Sample code for this would be:
Connect-ViServer vcenter.mydomain.com
$datastore = Get-Datastore mydatastore
New-PSDrive -Location $datastore -Name myds -PSProvider VimDatastore -Root '\'
At this point, you can access the datastore as any other drive at myds:\, so to rename a file on the datastore, you could do something like:
Rename-Item -Path myds:\ISOs\myimage.iso -NewName mynewimage.iso
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