Is there a way to monitor quality of service for a website that is totally run in Flash?
Yes, but it requires understanding what resources are potentially pulled in by the flash component.
My suggestion is this:
This gives you a general availability on the entire website driven by flash.
If you want to measure response times on particular components over time, you can add individual "html/asp" monitors for those resources you care about.
Alternatively, you can use the same technique - insist on "GET" instead of "HEAD" with a subset of those resources - and have the load time recorded in the database & reports.