It wouldn't be hard to fix this for the top bar only with a user script; in fact, the following code will do it:
$('.topbar img.avatar-me[src^="https://i.stack.imgur.com/"]').attr(
'src', function (i,v) { return v.replace(/\?.*$/, '') }
);
This is slightly wasteful of bandwidth, since it uses your full-size avatar image, but it shouldn't make that much difference as it will only affect you, and your browser will cache the image anyway.
However, I'm not sure how much point there would be in doing this, since the change would only be visible to you, and only in the top bar. What we really need is a proper fix to Profile picture cannot have transparent background, and that's something that Imgur needs to do — it's not something SE can fix on their own (other than by switching to another image host).
Edit: I went ahead and included the fix above into version 1.4 of the Stack Overflow Unofficial Patch, a collection of client-side CSS and JavaScript fixes packaged as a GreaseMonkey-style user script. If you're using a compatible browser (Firefox, Chrome, Opera, possibly Safari and others, but unfortunately not IE or most mobile browsers), feel free give it a try!