So far I've not been able to find a way to do this so I expect it can't be done, but does anybody know a way of querying either API v1 or v2 to find out how many times a particular custom tag has been used?
Tag's API returns it weight which represents the usage count of that tag.
Try to hit this API
Here you can see how the response will look like https://prnt.sc/kl2ow6
Thanks @VikasB but I think that's just for use with standard front end tags, not the Moderator custom tags... as in these ones: