I was in a similar situation with my two. Muffs was about 5 mths when I adopted Abby, who was only 10 weeks at the time. Muffs was (and still is) a very timid cat, whereas Abby was a normal kitten who just wanted to play. I separated them initially. When I tried to put them together, Abby constantly wanted to chase/play, but Muffs misinterpreted her "advances" and would run and hide. If Muffs couldn't run fast enough or if Abby kept chasing, a fight would ensue. After about 2 months of getting nowhere fast, I finally hired an animal behaviorist to help me deal with the introduction.
He said the problem was Muffs...the fact that she was so timid and was terrified of Abby. I suspect you have a similar problem with Punkin. I can tell you what the behaviorist advised me to do. I don't know whether it will or won't work in your case. Fortunately, it worked in my case.
First, keep them separated as you are with the screen door in the bedroom. Jack should be in the bedroom, and Punkin should have the run of the house. Put all food, water, and comfort objects (toys, etc.) within 2 ft of the door on either side (or as close as possible). Have all “good things” in life happen within 2 ft of the door (all attention, wet and dry food, treats, catnip, playtime, cat grass, grooming, etc…) in each other’s company. Try to have at least 10 "good things" happen each day (2 or 3 meals, several play sessions, a few treats, etc.). Switch the cats for one hour each evening. During this time, just let Jack wander around freely and let Punkin in the bedroom. They need not be by the screen door during this time. Monitor their behavior. Once you start to see a reduction in reactivity -- Punkin no longer acting frightened, the two cats starting to interact through the screen (playing footsies, etc.), then conduct a supervised 10 minute play session each day with the two of them together in an open area. After the session, put Jack back in his room and continue with "good things happening by the door". Over time, gradually increase the length or number of the supervised play sessions where they're together.
I wouldn’t use the carrier, since the cat in the carrier will be too confined. If Jack is in the carrier, he’ll feel frustrated. If Punkin is in the carrier, she’ll likely be terrified. So, the bedroom is better, since both cats have an “escape route” away from the door if they feel the need.
I was quite skeptical that any of this would work, but after a week I started to see some improvement. After about 2 or 3 weeks, Muffs and Abby began to play footsies under the screen and they started to touch noses through the screen. I then started to allow them out together for short, supervised play sessions. At that time, I had to constantly distract Abby (with toys, a laser light, etc.) to ensure she didn’t chase Muffs. It took another 3 weeks or so before they could be out together all day with me still home to keep a general eye on them. It took another month before I left them alone together when I was at work.
All in all, it was about 4 months from the time I adopted Abby to the point where I could leave them together. It was a lot of work...constantly playing with them on either side of the door, etc. But, on the plus side, they are now best friends; they cuddle and groom each other and you rarely see one without the other. I hope you have the same good luck.
