Hey Pry,
I have never done this, but I certainly don't think it's a bad idea but I think its the wrong forum for game. The circumstance I see using linkedIn working the best is in a situation like this:
1) you have ties to the girl already AKA you have met in a formal setting before whether that be school or work and you are "connected" already on linked in.
2) you keep your linked in interaction to a bare minimum, and basically just use it as a reaching out point to hop to another form of communication, whether that be FB, insta, or preferably a phone #.
Note this is not field tested, but this is what I would start with when testing.
"Hey
name, looks like you are (doing well for yourself | still hanging out at
the company you both used to work at)

how are things going?"
I would wait for a response and if you get one - then I would go straight for the date/number close. So it's a pretty safe message. After her response is where I would need some testing and it kind of depends on how you knew the girl. The less you know her, the more banter I think would be needed before closing.
"(assuming she asks how you are doing) I am doing great!
New job is treating me awesome."
"If you promise not to tell HR

I think it would be cool to catch up over a drink sometime soon, any interest, you just gotta promise not to complain about work the entire time?"
All that being said -- I think its not the right platform and probably won't net you too much results, BUT its worth a shot if you had a hot co worker that you could never chat up at work in a previous job. But its something that could probably get you reported to linkedIn if you abuse it and just hit on random women. It would require a TON of tact to make this work consistently. But for certain situations, I see nothing wrong with it.
If you do this, please post some FRs about your interactions, I am really curious.
Cheers,
Radeng