Maybe a simple spritesheet player would be a decent tool to have. Even as an app for a secondary device/screen.
That's a really good idea - I don't need to preview it in-game all the time, I just want to preview the spritesheet running as I save. I hadn't actually thought of it that way. I'll poke around online and see if there's anything closer to that.
Edit: So I ended up finding FlxSpriter. You give it a PSD file, setup some frames in XML and then you can play your animation on loop. It even live reloads! It took a bit of effort to get it running though (it depends on a few libraries that the readme doesn't mention) and the fact that you have to manually update an xml file is a bit of a pain. But, this is much better than my current system. I also found Spritesheet Preview which has a much nicer interface (no XML files) but it doesn't support live reloading and can't read PSD files.