Can SSI trigger the ErrorDocument setting?

I'm using an SSI "include" directive in some files. For example:

<!--#include virtual="/include/dummy.html" -->

However, some of the included paths don't actually exist, so Apache
shows the standard SSI error message instead:

[an error occurred while processing this directive]

Is there any configuration trick that will make Apache follow the
ErrorDocument directive instead of displaying the SSI error? I realize
I can configure the standard SSI error message to say something else,
but I what I want is to invoke my standard ErrorDocument page/script
which will perform some processing and then spit out an error message.

Probably not possible, but clue me in if there is a way!