Thanks for the ideas! I think a crate on the bed is a good plan. I have a day bed with bars around 3 sides, so a crate would sit perfectly at the end of the bed. I've been doing some training with Mimi, and I think we're making progress. Right now my mum and I are on the sofa with Mimi and one of the cats, Kim. When I first sat down I had to apologise to Kim because I sat on him! He sat on my lap while I apologised and cuddled him. Mimi asked to come up on the sofa by scratching my legs, but I ignored her while I paid attention to Kim. As soon as she stopped and looked away I praised her for sharing, and a few moments later picked her up and put her on my lap. She walked over me to make a nest on the sofa behind me, ignoring Kim who decided to sit on my mum's lap. We've been in the same position for half an hour or so.
What I've been doing on other occasions, is to cuddle and pay attention to the cats while praising Mimi but not looking at her. When she looks away from me I praise her much more and pay attention to her for a bit. While her being possessive wasn't a constant thing, I still feel fairly optimistic! She is rather possessive of her food though (with the cats, not me), so her and new dog will be fed separately.
A couple of hours later, and the other cat just tried to sit on my knee - Mimi didn't like that and chased her off! So I immediately put her on the floor and said 'no'. She asked to come back up but I ignored her, and waited for her to walk away. When she did, I called her and put her back on the sofa. Hopefully she'll get the message!