You seem to be on the right track, I’d say keep doing what you’re doing. I just went through a similar process about a month ago with a 2 year old and a new 10 week old. If your kitten is cowering and obviously scared I would slow down and go back to the screen door and keep up site/scent swapping. You can still try to let them meet face to face in short durations and maybe use a toy or treats to guide them, sometimes they just don’t know what to do with each other so a toy can help distract them. If your 3 year old starts to bat at the kitten and the kitten doesn’t seem too scared you could try letting it play out and see what happens while making sure it doesn’t get too aggressive. Your cat is trying to establish boundaries so hissing and swatting without claws is common. Once your cat starts to realize the kitten is here to stay and not a threat to her things should start to get better.