IIS 7.5 One Application Pool works, a second one doesn’t work

I had the same issue as well. As per this page, the ApplicationPoolIdentity is supposed to access network resources as the machine account (DOMAIN\COMPUTERNAME$), but it was using NT AUTHORITY\ANONYMOUS LOGON to access SQL server instead.

I was able to use this hotfix to get ApplicationPoolIdentity working as the doc says it should. This hotfix doesn't specifically describe a solution for accessing network resources as NT AUTHORITY\ANONYMOUS LOGON, but it's related to the computer password changing.

This stackoverflow thread helped me find this solution:

Other useful tips for debugging ASP.NET identity and SQL identity

Since SQL integrated auth is related to the thread identity, it's useful to be able to view your user identity, thread identity (which changes depending on impersonation), and the SQL server login. This ASPX snippet displays all three:

<%@ Import Namespace="System.Security.Principal" %>

<div>ASP.NET Request User identity: <%= User.Identity.Name %></div>
<div>Current thread/process WindowsIdentity: <%= WindowsIdentity.GetCurrent(false).Name %></div>
<asp:SqlDataSource ID="CurrentUserData" runat="server" ConnectionString="<%$ ConnectionStrings:ConnID %>"
           SelectCommand="select SYSTEM_USER" />
<asp:DetailsView DataSourceID="CurrentUserData" runat="server" />

In IIS you cannot just route request from one application to another. Applications are isolated, that is why you are getting 403 error.

You can proxy requests using either ISAPI_Rewrite, Ape or ARR - it does not matter because request will be passed to another application using local HTTP request anyway. This solution is quite stable, but you will lose some performance.

Redirect is probably not an option here, because it will generate two request to the server anyway, but since request will be generated by the user with slow connection performance may drop significantly.