Finally I found the problem. I added location element to the config in %WINDIR%\System32\inetsrv\config\applicationHost.config with the headers I needed and headers start appearing in the response
<configuration>
...
<location path="MyWebsite">
<system.webServer>
<httpProtocol>
<customHeaders>
<add name="Access-Control-Allow-Origin" value="*" />
<add name="Access-Control-Allow-Headers" value="Content-Type" />
<add name="Access-Control-Allow-Methods" value="GET,PUT,POST,DELETE,HEAD,OPTIONS" />
<add name="Access-Control-Expose-Headers" value="*" />
<add name="Access-Control-Request-Headers" value="*" />
<add name="Access-Control-Request-Method" value="GET,PUT,POST,DELETE,HEAD,OPTIONS" />
<add name="X-Frame-Options" value="DENY" />
</customHeaders>
</httpProtocol>
</system.webServer>
</location>
...
</configuration>