My girlfriend ALWAYS brags to me about guys hitting on her. In shopping centres, at clubs, or at bars.

So, what's the best response? I know, to appear unaffected and to not give a shit. But what about taking it a step further and getting girls to flirt with me? The only issue is that I have a much smaller social circle, and I've tended to ignore girls hitting on me (the very few that have) as I prefer loyalty over anything that can be misinterpreted as unfaithful. So, would it be wrong for me to start sarging but only with the intent of friendzoning girls so that I can say "hey check out this girl who added me on Facebook, she was totally all over me today"?

Your girlfriend seems to have insecurity issues, and a desperate need for attention. She WANTS a reaction from you, but not just any reaction. She wants a reaction that shows you're insecure. Don't feed the troll.
By completely ignoring what she says, it could potentially be interpreted as you being bothered by it but pretending to not care with your head in the sand. It's also an abnormal reaction and she'll just keep doing more and more outrageous shit and enjoy seeing you bite your tongue.
DEFINITELY don't do the Facebook trick you mentioned. That will basically let her know that she won and you're worried. I could suggest that you do what I've done in the past, but it may seem out of character for you and backfire. A girl I dated (not the last one you saw me mention, but before) used to tell me about guys that hit on her (I know the guys around her. They are absolute nobodies, like someone mentioned on a nearby thread, if she wanted to cheat on me with one of them, I'd drive her over there with all her stuff and laugh all the way back). My response was always condescending and generally douchy. eg: "daww that's adorable" and "shoulda picked him.. I don't think I'll ever measure up". Are you normally cocky with her? If not, you should slowly become that way. Don't even deny it. I straight up told my gf that I'm a cocky asshole... She denied it and said I'm very modest and sweet...(I'm not). Honestly, just give less fucks about her, the other guys, and having to be with someone in general, and you're 80% of the way there.
If you want to make her worried and jealous, be subtle. VERY subtle. She'll pick up on it from your subconscious actions/words. Go, flirt with some HBs, see them as just friends, LITERALLY, and make sure it doesn't look like you're just parading them around in front of your GF to make her jealous. Mention them as just friends, never mention their looks in a way that makes you seem interested (I'm PMing you a couple strategies off the record). Hell, see if you can get another HB interested in you. Don't cheat, but the knowledge of having attractive options alone, will give you the confidence, and the lack of fucks given to deal with your attention seeking problem girl.

Just know, that unless you just started dating, the behavior you're seeing is a red flag, and you should think twice before getting more involved with her.
Good luck my friend.

Well yes she's definitely insecure and that MAY be interpreted as my fault since she thinks I don't tell her how beautiful I think she is (though in reality I tell her every other day and I've already made a decent effort to remind her more frequently). I just find it disrespectful for her to freely tell me how much guys hit on her and I want to up the ante a bit. I already play the don't-give-a-shit card and say things like "aw you should have made out with him."

She doesn't really play games, and if she does she's bad at them so I don't find it necessary to go full-out and deploy nasty tactics. But it's always nice to have an extra set of weapons on my holster if it becomes necessary. I'll try implementing some of the strategies you suggested though.

Keep us updated on the HB that dumped you should you hear back from her.

Lol I'll let you know about the last HB if I hear from her. If not, my life won't be any worse.
As for the compliments. No matter how many times you tell her she's beautiful, it will never be enough. That's my moneyback guarantee to you. Don't supplicate, and stay more judicial with your use of compliments. The girls get spoiled, take them for granted, and expect more.