The "opt-out" setting seems to be stored server side on a per-user base, so there is no easy way you could achieve what you want. After all, even if you do access the site from two different browser on two different machines your user profile is still the same and so is any hat related configuration you made.
You could still try some workarounds in order to stop the Winter Bash script being loaded in your browser. I don't know a lot about Opera 12 on linux, but two main options come to mind:
try and block content from the domain "cdn-prom.sstatic.net" either using an origin based resource filter (something like uMatrix on Chrome) or some of your browser built-in features (if available). As a reference I use Firefox and if I configure NoScript to block script from cdn-prom.sstatic.net hat disappears completely on meta (I think that would also apply to chat).
hats aren't supported on Internet Explorer. While you can't really switch your browser, maybe you can trick the servers into believing you are using IE. Again, I don't know how they implemented the "No Hats for IE" feature, but maybe it would be worth trying to switch your browser user agent string to one used by Internet Explorer (warning: this could potentially break the rest of the page if they are using IE specific fixes)
I think those are the only available options. I think your best bet is still blocking the "cdn-prom.sstatic.net" domain.