In my WebAPI application, I faced similar issue of 405 method not allowed web api and this issue; Asp.NET Web API - 405 - HTTP verb used to access this page is not allowed - how to set handler mappings
I had to disable WebDAVModule in my web.config like below;
<modules>
<remove name="WebDAVModule"/>
</modules>
<handlers>
<remove name="WebDAV"/>
</handlers>
I do not depend explicitly to this module, but I am just wondering that would disabling this module could create any significant performance/security problems? I checked documentation below, but could not find anything specific to it.