Don't do this...
The dating site charges a high fee. I'm not super comfortable paying to message just one person. Would it be considered harassment if I messaged them directly at work? (I'd find this creepy myself so probably yes?). If it is I'd just pay the fee.
Your gut instinct is good on this one. It would be extremely inappropriate to bring up the fact she's on the dating site in conversation - let alone actually acting on it to ask her out. She's using the dating site outside of work, not during.
More simply, even if you got the information that she's potentially looking for a partner via the dating site; sending anything like this during work, especially via work communication, is completely unsolicited and inappropriate.
Would it be considered harassment and a risk to my career if I approached her on the dating website? Both of us are aware the other works at the company
No and Yes.
Would it be considered harassment? If you work nowhere near her, have no ties to her or her department, and were careful to respect her boundaries (as you should be anyway); then it'd be unlikely to escalate into a harassment claim (not impossible though).
Is it a good idea? Not really. You might not work together right now, but you don't know where you'll be (or where she'll be) in the next year. Even if you're completely unfazed by working with somebody you wanted to date - she might not be.
I'd strongly recommend you treat your current workplace as the biggest compatibility red-flag you can. If in all other aspects you think you might be interested in her; hold the fact you work together against it - it's not worth the risk.
It's not romantic to consider; but the reality is she is no more likely to be "the one" than the many other potential romances you might pursue. All good relationships are built over time, without ever being a perfect fit. Ignoring that you ever saw her profile is not going to hurt your chance to have a romantically fulfilling life - but it will do a lot to protect your career.
In my case I feel that doesn't apply - we're not even in the same building, and it's a huge company.- The fact that you've "seen her around before" makes this comment irrelevant. The mere possibility of running into someone she declined on a dating site may create a hostile work environment for her. That is not fair to her regardless of the size of the company.