There are two filters to check. One is to allow Flash Video .flv and .f4v and the other is for Flash Animation .swf. There is a warning note with the latter to be used in trusted texts as it has potential security issues. See Moodle docs: http://docs.moodle.org/21/en/admin/setting/filtersettingfiltermediaplugin
Your Admin may be reluctant to turn the .swf plugin on.
I forwarded this to the admin and he said that he did both of these. With certain browsers the window asking what to do with the file doesn't even pop-up, it just keeps thinking trying to do something. I've tried rendering the Camtasia movies as other files, but that doesn't seem to work either. Any other ideas?
Don't upload the swf. Camtasia doesn't know that you are uploading to moodle.
Moodle has it's own built in player called FlowPlayer. Simply find the flv/mp4 file next to that swf and upload that instead.
You will have to enable the multimedia pluggin for it to see the flv but it will work fine.