How can I disable group event notifications from popping up?

Simple, don't join any groups :)

And if you've already made that mistake, you can always leave the Groups in question...

Failing that, I have no idea what to suggest except try to ignore the notifications...

That's right, you can't. Same with all of us we have the same problem. maybe you don't know this but the feature is being asked since at least 2009, I've found and commented on forums (yes, even oficial ones) and not a damn care in the world about actually implementing this.
The way I see it, you have 2 choices, nay make it 3:

• Delete the groups that annoy you with popup announcements (actually that isn't what bothers me/us; it's the fact that it fucking takes you out of the game you're playing! that's it, stealing focus.
• Ask as nicely and polite you can, to the owners of the group to calm down about the spammy announcements. Then be banned expulsed from the group. No words or good byes, just a blunt kick in the teeth..
• Deal with it, paciently curse all the way up of the spammer, from his ancestors to his progenie, close and then resume playing, that is IF your game didn't crashed because of the sudden quit-to-desktop-annoyance. Oh yeah it happened to me, that's how I know all this. Been there done that approach.

I'm not 100% sure it works, but when playing, you can just click offline connection, you won't get chats from your friends or giveaway alerts either. Thanks Valve!

Has been a problem for many years, Valve have not shown any signs of putting something in to deal with this

Only way is to go to friends and choose offline as your status.

You can't, enjoy them as much as I enjoy have to censor them on my videos.