@Alexander's answer put me on the right track. Had to add the following to get DELETE/PUT handled by ASP.NET:
<system.webServer>
<modules runAllManagedModulesForAllRequests="false"/>
<handlers>
<remove name="ExtensionlessUrl-Integrated-4.0" />
<add name="ExtensionlessUrl-Integrated-4.0"
path="*."
verb="GET,HEAD,POST,DEBUG,DELETE,PUT"
type="System.Web.Handlers.TransferRequestHandler"
preCondition="integratedMode,runtimeVersionv4.0" />
</handlers>
</system.webServer>
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