@Alex - So, I see that while the code has now changed on the live site, the effect is still happening, especially when there's a very slow loading flag on the war page (like for me right now - Bamika's image takes 7 seconds to load, wtf). Given that I proposed the prospective fix and it's still not working right, I felt somewhat obligated to investigate further
After dabbling around, I believe I've found the problem, though I'm not 100% certain on how best to fix it. I suspect the issue is Rocket Loader, which is injected by Cloudflare to asynchronously defer various blobs of Javascript for the purpose of making the page load faster. Apparently it works by replacing script type attributes with a prefixed version thereof (for example, prefix "bce348fb34a3edbcc529509b-"), then later going back and loading them on the side. Quite an insane piece of tech, honestly. If that is indeed the case, then why it's broken is clear - that piece of javascript to bind the drop down actions isn't executed immediately with the load, but rather much later whenever Rocket Loader feels like it.
I can think of three potential ways of fixing it:
Add data-cfasync="false" to the script tag, as described here. The problem here is that the snippet requires jQuery, so the script tag for loading jQuery in the head section needs that attribute as well.
As above, except instead of also "synchronously" loading jQuery, replace that snippet with a vanilla js based version. Good ol' document.getElementsByClassName() should do the trick.
Turn off Rocket Loader entirely in the CloudFlare settings. This may fix other potentially wonky places as well, though it may make the site "feel" slower (or faster, depending on how the browser rather than CloudFlare decides to schedule things like Bamika's insanely slow image.
*phew*.
While I'm here, a question: Would you mind if I rig up a piece of Greasemonkey code to replace that dropdown with individual buttons and/or keybinds?