The DbContext is not thread safe so you should not use a singleton DbContext in an ASP.NET web app. A thread per request is created and you'll start getting exceptions as soon as multiple users begin using your application.
It sounds like you're wrapping the cache in another class that has its own DbContext. Why not just have that class take a DbContext in its constructor? Also, is there a reason you aren't using HttpContext.Current.Cache (Cache class)? That is the preferred way to Cache in ASP.NET (If you aren't using a 3rd party cache) and it will have the lifetime of the app domain. I'm fairly sure this solves all of your singleton and 2 DbContext issues.
According to Microsoft
The MemoryCache class is similar to the ASP.NET Cache class. The MemoryCache class has many properties and methods for accessing the cache that will be familiar to you if you have used the ASP.NET Cache class. The main differences between the Cache and MemoryCache classes are that the MemoryCache class has been changed to make it usable by .NET Framework applications that are not ASP.NET applications. For example, the MemoryCache class has no dependencies on the System.Web assembly. Another difference is that you can create multiple instances of the MemoryCache class for use in the same application and in the same AppDomain instance.
