Your post has a lot of red flags, not just about your son but about your own fears here. You "want my boy to be happy." You emphasize that you like the girl and you like the girl's mother. You "hate" to not give them "any space" despite your admission that they are using the "space" you give them to have a physical relationship you know is going to lead to more and more.
Please try to read your own post -- especially the last paragraph -- as if you were reading a stranger's post. Can you step back and be objective enough to see that you seem to be afraid you will anger and alienate your son if you do the parental thing and tell him, no more "space" alone?
Can you set aside the fact you like the girl, like the mom, and fear your son? You're trying to keep everyone else happy here -- your son first, the girl's nice mom, the girl too (yes, she has a mental health issue if she has diagnosed bipolar disorder, but that is NOT a reason to treat this situation with kid gloves--YOU are not responsible for her being upset if they can't date)....But happiness sometimes isn't safe or smart, and a kid has to be unhappy for a time in order to be protected from his or her own very dumb choices.
I agree with B, who replied below: Your son has too much time on his hands. Does he have any activities? He should, and if he only has one and it's leaving him this much time -- he needs to be doing more. Activities, clubs, church if that's your thing, chores (be sure to reward any extra chores, so he doesn't see them as punishment). If these woodland hickey sessions (and yes, more is surely going on, or will any second now) are taking place on weekends: Time for family day trips to whatever interesting stuff is near you. More family time, more homework time.
If you tell him the girl is forbidden, he will only see her as "forbidden fruit" and will probably both defend her (because he'll think you are attacking her) and will sneak around. At least for now you know that they're going off alone, and yes, it's good that he does talk to you. But I'd start tapering off "dates." If they see each other, they do it in public places with a parent nearby, and at the same time I would -- without telling him so directly -- just start ensuring he is far too busy to have time to slip out to the woods, period. No more merely hanging out at his house or hers, and if he has friends who would let them meet at the friends' houses -- nope, you need to watch out for that, too.
At their ages, neither is driving, right? So they can't go off in a car unless an adult takes them. Use that fact to your advantage, mom. Only public dates, no more than X times a month, and you drive them (and I'd stick close by if it were me). Then you start tapering the dates off: "We're going to X for the weekend as a family" when he asks if he can see her.
Remember that if you say, "This is keeping you busy so you won't get Girlfriend pregnant" he's going to buck it and probably sneak. Just make it happen and do not make it about her, do not badmouth her in any way, just ensure your family is busier and busier.
Summer is coming soon. He should have a LOT to occupy him so they aren't hanging out. Summer can be a good and natural period for a break in a friendship or dating relationship that is getting too intense; use that time.
One big red flag: You say he gets angry when you question this. How angry? Does he act out? Does he get physically violent? Look at how he expresses his anger. That may need a lot of work and may have roots that aren't related to this girl but to his being unable to handle being told "no." Does he get angry when you say no to other things?
You don't mention whether his father is in the picture or whether he has another older male role model. It could help hugely if there is a father, uncle, coach, older guy he likes and trusts who could talk to him very bluntly about sex and why 14 is WAY too young, and how if they're "just necking" it IS going to lead to sex. He also needs to hear that condoms do not work 100 percent of the time, but kids are 100 percent commitment. Is there a man in the picture for him?