I've been experimenting a bit more with FunctionFlip, and it is easily the best solution. I want to go ahead and add some tips.
First, use FunctionFlip to disable using F7, F8, and F9 as media keys.
EDIT: Also be sure to enable the option to start FunctionFlip on login.
Next, when editing VLC's hotkeys, you'll find that F7 and F8 are already bound to functions for playing a bookmark. You'll need to clear those out first. Then you can reassign F8 to Play/Pause, F7 to Short backwards jump, and F9 to Short forward jump. Alternatively, you could use the medium or long jump options.
Once you've saved your new hotkeys, you'll need to exit VLC and restart it to take effect. Now it'll work perfectly.
In iTunes you no longer will be able to use those media keys without also pressing the FN button. But you can get around this by setting custom keyboard shortcut preferences. Go to:
System Preferences > Keyboard > Keyboard Shortcuts
Click the + sign and find/add iTunes.
Then for the Menu Title, enter Previous, and use F7 for the Keyboard Shortcut.
Repeat with Next as the Menu Title, F9 as the Keyboard Shortcut.
For some reason, setting Play to use F8 doesn't work, but you can leave it be and just use the spacebar. It'll work just the same.