Apple refuses to allow Flash Player on their device. Adobe AIR actually compiles the .swf into native code when deploying applications to that device.
Also, I am pretty sure even on Android the Flash Player is being retired because if you update to certain Android versions you are warned that you will no longer be able to install Flash Player. I think Adobe is moving 100% to Adobe AIR on mobile. Maybe in the future it will be able to play .swf files without a plugin? But for now, I am not sure there is a way to view a Flash file on all devices.
I just don't think you are going to be able to do what you are trying to do but I would love to be proved wrong.
HTML5 is not an approved standard, it's not always the answer unfortunately. If the OPs whole site is HTML4/XHTML/XML/etc, recoding the entire site just to play video may not be feasible. Not to mention limited support for HTML5 in older HTML editing products.
You're using the built-in component in Dreamweaver so you should consider something like Flowplayer which supports MP4 so your videos will appear on iOS devices as well as support fallback to FLV. It's a free component-like video player for sites that you can customize. You can find Flowplayer here: http://flowplayer.org/
If you're willing to make the site HTML5 then here's how easy it is to add a HTML5 video to the site: