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Would it be feasible to change the name of the resource in this case? When a resource's name is changed, it is deleted and recreated (as a side-effect of being a completely different resource as far as Pulumi is concerned).

No, the name should be kept the same. I guess what I'm looking for is a "Replace Instead of Update" resource option, similar to the "Delete before replace"

But to be honest I don't have a use case that justifies this.

The name is just a property like any other; if you come across a use case where this behaviour is important, then a little cheat like changing the case of the name string just to get a delete to happen might be ok? After all, the name is used for nothing except Pulumi's internal referencing.