I'm not familiar with PHPWebsite, but I have some experience working with a similar "community center" template-driven web page authoring software (that particular system uses ASP scripts to create webpages). If my guess about PHPWebsite is correct, then what you're doing is uploading content
to PHPWebsite, and PHPWebsite probably automatically creates a page around your content using a template that you selected for the page.
In the system I'm familiar with, I can enter content into text areas in the template, and I have limited ability to insert CSS and HTML code. You may be able to embed the HTML portion of the code that Flash RSS generates, but you will have to ask your system administrator where exactly in the server you can upload the additional files generated by Flash RSS. It's also very possible that you many not be allowed to upload additional files, since the very point of software such as PHPWebsite and other community center software is to insulate content creators from such "worries!"
But contact those Powers That Be that operate your PHPWebsite system. That would be the best bet. Good luck!
Mostly just lurking these days ...