The reported behavior is caused by the fact that the BingMapProvider derives from the Freezable
class. When you define a freezable object in a DataTemplate, ControlTemplate or any resource, the WPF framework automatically freezes it and it cannot be further modified. This is why when you define RadMap into the ControlTemplate of the custom control the exception is thrown when the provider's Mode property is changed.
The OpenStreeMap and ArcGIS providers are also freezable object, but their mechanism of changing the view mode is different and this exception doesn't occur.
In order to resolve this for the BingMapProvider you can create a custom provider that derives from the bing map and override its FreezeCore() method, then cancel the freezing. Here is an example:
MyBingProvider : BingMapProvider
Please try this approach and let me know if it works for you.