New cat, sweet to people, not to resident cats

Our home recently added a new 3 year old female cat to an established "colony" of a 1 year male, and another 3 year female. The new cat is the focus of this post, so some background there. Her name is Jaina She is an orange tabby who recently had kittens. Not certain how long ago exactly. She was likely (unclear) an outdoor/indoor cat before. We picked her up from the shelter just after she was spayed. Jaina endeared herself to us very easily as she was so sweet. She's a lap cat and loves company. But there was little information about how she got along with other cats. We were suckered in anyway and brought her home.

For almost a week we kept the cats separate. With occasional sightings through the door. We swapped scents and places a few times, and let them see through the door on purpose later. There was no apparent tension. A little wary maybe. And we found that Jaina was getting more and more stressed being alone despite quite a bit of time spent in her room with her. So we let her out. Jaina started hissing at our other two cats, Luke and Mara Jade. Not on first sight. First when Luke got the courage to come of sniffing, very slowly. Of course they were separated again. We got a Feliway collar for Jaina. It hasn't helped much. She started digging up the carpet under the door trying to get to us. On top of that we took her for her first vet visit and they found a uteran granuloma. She got a shot for it and we'll take her back next week to get it checked out. This certainly hasn't helped Jaina's stress.

A few days later we tried another supervised introduction. Luke and Mara were a little more cautious this time. Still, they would be fine with Jaina's presence. Until Jaina would hiss at them. Distractions worked at first. It's important to note that Luke and Mara have never shown any aggression. Rather, they have each multiple times tried to cautiously and gently approach Jaina only to be rebuked. Then later, during another attempt, Jaina ran toward Mara and hissed at her. A very aggressive, though not physical, move. Jaina even growled and hissed when she heard Mara trying to eat. I was holding her, but she used some claws, got out, and rushed toward Mara. Since then Luke and Mara have hid whenever they see Jaina.

We've sequestered Jaina again. In the basement. It's a little colder down there, and noisy. That's unfortunate. We'd like to move her to a more comfortable room but she is tearing the carpet there. And she's getting more stressed. Possibly from being alone. We can only devote so much time to being with her. For now, we're starting over with scent swapping and site swapping. We've got a plan to make a barrier they can eat on each side of. 

While Jaina is kept separate Luke and Mara are themselves. Great cats. Friendly as can be to each other. Jaina is wonderful with just us. Put them together and it's a disaster. So we can't let Jaina out like that, and that leaves her stressed and alone. We try playing with Jaina. She isn't very into it most of the time. She mostly wants company, but we can't always give that to her.

The real questions are:

Are we being fair to all the cats?

Is a peaceful solution likely? 

If there can be peace, what are the best tactics for this situation?

How can we reduce Jaina's stress in the meantime?

It can take months for cats to adjust to each other. I've found that the younger the newcomer is the quicker they are accepted but once you get past 8-9 months it gets harder for resident kitties to accept the newcomers and vice versa. I personally have a very strict one female per household policy because I had a nightmare situation when I was younger (it was actually no females but Rocket was a surprise girl). But you are right that the basement confinement is probably not helping the situation. 

You don't mention if everyone is fixed, I am assuming they are, but that can make a big difference if they aren't. You could try the feliway sprays and room diffusers, some people have reported good results with that and that would be affecting all three cats instead of just the one with the collar. I am not a fan of the collars because they can be dangerous and I would think that other scent overpowers the feliway scent. There are also essential oil mixes from Jackson Galaxy; since they are on the cats themselves you need to be careful to watch for allergic reactions, hair loss and rashes. 

I'd say to keep trying to bring them together and distract with toys or treats when they start hissing. You can also do the method of feeding them with a barrier between them so they start associating smell of each other with good things and over time slowly remove the barrier. There is also the method of taking a towel or blanket that you exchange between the two groups so their scents get intermingled. You could try letting the new one into the house while the residents are allowed to explore the basement so everyone smells each other and explores the other's area. Remember to give each way a few weeks to work before moving onto something else. There are different options out there and I wouldn't say it is hopeless yet. But I would give yourself a point where you say it just isn't the right home, probably a date deadline or after you've tried the different methods or if the tension starts to negatively affect their health, and see if the rescue group will take her back; there is no shame in knowing you tried everything and it just wasn't the place for her. 

Try watching some episodes of My Cat From Hell, he deals with cats not getting along in a lot of episodes and has a lot more experience and insight into methods to get them coexisting. I've mostly had kitten introductions which tend to be easier. My females problems didn't start until they had both been living together for two years and were triggered by my nieces birth so it was a different situation.