I don't think she is pregnant, and it doesn't sound like she thinks she's pregnant.
It doesn't sound like if you were to end the friendship she would claim to be pregnant "just to keep you around," it sounds like she is very clearly telling you she is not pregnant. (But if she were to claim pregnancy, a pregnancy test and a DNA test solve that problem, you don't need to stay around.)
You are casting yourself as the victim, and finding ways to put her down after having slept with her. This is an ungentlemanly thing to do. I get that you're afraid of an unexpected pregnancy. But it is not necessary to say she is "emotionally manipulative" and "found a way to talk romantically" to you (you can't be manipulated without your cooperation). You say she is bipolar and went to a mental hospital (presumably seeking help or healing, which should never be used by anyone to portray the person who sought medical help for such issues in a bad light). These are her problems and her cross to bear, but certainly she never told you about them so you could use them to justify your critique of her that springs from a fear that she is pregnant.
The next person you meet who flatters you and is flirty, don't hop in the sack just because she's sending out signals. Learn to use condoms properly. Only sleep with a partner who is on the Pill. Don't add to your partner's problems by being unwilling to ask any questions about contraception until it's too late.
When you tell her you want to end the connection (and don't avoid her, simply tell her) explain that it is because you don't want to lapse into a physical relationship again. Don't imply that she is being an evil manipulator of poor little you. Stand tall, say thank you for the good moments, and say you're sorry but you can't be friends any more because you (not she) are unable to hold a line at friendship even though that's all you want. This kind of honesty might sound very challenging, but it will be better for her and for you, in the long run. Everyone who has slept with someone else deserves at least a clean explanation of why it's over, when it is.
Good luck. I suppose you'll think this is harsh, but I'm trying to get you ready to be a great lover whom all future sex partners will be glad to have known, not a nervous guy who blames things on his partner. You can do this, and it will be to the good for you.