I'm going to have to respectfully disagree with the sentiment that having a threesome would be a good idea. I would NOT recommend initiating a threesome unless that's something that you're really into doing. If you're already uncomfortable being around your partner's other significant other (to be herein shorted to OSO) being in an intimate situation with both of them present is likely not going to help things, on the contrary, it's likely to bring up even more insecurities and drama, and depending on the personalities involved it may make it more difficult to communicate effectively. (Or, at least has been my experience).
That said, both you and the OSO have perfectly valid feelings about this. Some people are okay with giving their blessing to the other relationship but prefer to keep their distance because it's too uncomfortable to see their partner with someone else. Other people, like the OSO, have the need to be friendly and occasionally included in things with their partner's OSO because otherwise their imagination runs rampant on all the fun things they must be missing out on.
It's hard to come out with a blanket answer to the question of "Will this work if I'm not friends with her?" because quite frankly, it depends.... If being friends is what she needs to feel comfortable with this arrangement, and you feel that you can't be around her, chances are things aren't going to work out or are going to be very drama prone.
The key, I think, is working toward some sort of compromise.
First, I think it's worthwhile to sit down and do some thinking and try to figure out what about seeing them together really bothers you. How I do this is to write down everything that's bothering me about a situation no matter how significant or insignificant it may seem. Then, I go over my list (usually later, when i'm less emotional and thinking more logically) and figure out what I wrote which were legitimate concerns, which were just me being upset about things that really didn't matter in the grand scheme of things, and which were things that I needed to discuss with my partner for reassurance. Identifying why you feel the way you feel is a great exercise for personal growth and it helps you be more self-aware so you can communicate effectively with others.
Second, don't feel that you ought to be comfortable hanging out with both of them on the living room couch right away. Start slow with baby steps. You mentioned that sometimes when you find out that they're both going to an event, you avoid it but at other times you deal with your jealousy and uncomfortable feelings and go. This is a wonderful place to start. I'd try to make it a point to go to the events you want to go to, especially if she and your partner are going to be there. It'll be uncomfortable at first, I'm sure, but often times part of relieving anxiety around a situation is repeated exposure to it. Going anyway, and then coming home and examining why you're feeling uneasy about the things you do, and then going out and then trying again next time does tend to make things easier over time.
Once you are comfortable with going to events in public space where your partner and his OSO are present, all of you can re-evaluate what your needs are in regard to the situation at hand.
So, short answer... The appropriate thing to do in this situation is to use this as an opportunity to grow as a person.