HELP! Cat gone to neighbours

I have two cats: Tj (F, 10) and Mango (M, 3).

I got Tj (Calico domestic longhair) 10 years ago when I was only 10 years old. She is like one of my best friends and has helped me through countless stressful days. She is an outside cat so she would spend some of the day outside and then come inside and chill with me and sleep inside. We are quite close to our next door neighbours. They are an elderly couple and their adult son lives with them. A couple of years ago our neighbours actually told us that that sometimes Tj would go their house or garden and would want food, despite me feeding her regularly 😂. We found it really funny because it was like she was living double lives, trying to get food from everyone. Our neighbours are really good people and we have a good relationship with them. We even trusted them to look after Tj when we were on holiday. Eventually, they would even buy cat food to feed Tj because she would be around their house often. However, she would always mostly be at my house and sleep here every night. So despite getting food from other neighbours, she had always seen my house as her territory and definitely recognised my house as her home. Tj was incredibly territorial and would always fight with the other neighbourhood cats.

In 2021, my older brother wanted to get a kitten. He asked me if he could get the kitten but I argued with him and told him that I know that Tj would hate it and I did not want her to feel unwanted. I asked him not to get the kitten but after much persuasion, he ended up getting it anyway. So in 2021, a new kitten came. His name is Mango and is a British shorthair. We got him when he was only three months old. When he came, I researched how to introduce two cats and made a plan. Mango was kept in a separate room for a while and Tj didn’t know any different. After a couple of days, I started to introduce their smells to each other and gave them food on opposite sides of the same door. However, things ended up getting rushed and they eventually would come face to face. Tj, of course, reacted badly, rightfully so and exactly how I predicted. She would hiss at him but Mango was really just intrigued by her. Because Mango was an expensive cat, we made sure Mango was an inside cat. After a couple of weeks, Tj would come in and sleep inside as normal but Mango was far too enthusiastic and would run at her and jump on her. This carried on for a couple of months. At this point, Mango was not neutered and so I knew that because Tj was female he had a particular interest in her. One time, he even jumped on top of Tj and tried to bite her neck to try and do something inappropriate. Over time, Tj would come into the house less and less and it got to the point where she would only come in to eat. During this whole time, Tj of course was spending her time at our neighbours. My neighbours adult son took a particular interest in her and even gave her a new name. I heard her calling him Julian and was so angry. Eventually, after two years, it got to the point where Tj would not enter my house at all. But i still fed her everyday outside the front door. I knew that i become easily attached to pets, so when Mango came i became attached to him instantly. So giving him away was not really an option. But i have a specific bond with Tj and she has practically seen me grow up and so i want to make sure she feels welcome in my home because it’s her home as well. I eventually got Mango neutered, thinking that it would decrease his interest in her. But it changed nothing in his behaviour towards her.

Currently, this is how the situation is. Mango is now nearly 3 and the whole family likes him. He has still remained as an inside cat. Tj never enters the house unless I pick her up and put her in myself. The times that I have done this, most of the time she will refuse to eat and will go upstairs and stay under the bed until she has an opportunity to go outside. Or she will wait next to the front door and try to leave when someone opens it. She sometimes hisses at me as well which she never done before. Whenever she is inside, which is rare at this point, I make sure that Mango does not get near her. But he always knows that she is there and will sit on the opposite side of the door waiting for his opportunity to get to her. I still feed Tj outside whenever I see her. But the problem is that my neighbours have practically become her new home and it frustrates me so much. They still call her Julian even though they know her name is Tj. They have a bed for her inside and she sleeps there now. Whenever I have been at their house, she is always there lounging in the garden or the living room. Whenever the son comes back from work, i always see Tj running after him. They even said how they are glad their son has Tj because it’s good for him, even though she’s my cat!! Whenever they give us greeting cards, they sign it off with their names and add Julian at the end. It is so incredibly frustrating hearing them talk and act like Tj is their cat, even though I got her when I was 10 and she has been a part of my life ever since. Once my mum even jokingly said to them that I get annoyed when I hear her get called Julian and that I insist her name is Tj, so in the next card they gave us they signed it off with Julian with the name Tj in brackets .my neighbours are good people but it’s the way they act and talk about her that really irritates me. Especially their son because I know that he is mostly the one that is doing all these things for Tj and has basically assumed that he owns her now. Essentially, they see her as their cat now. I know that I can ask them to stop feeding her. But first all that wont guarantee that they will stop feeding her and second of all that does not solve the fact that Tj refuses to come inside my house. Tj is getting older now and she noticeably looks different. She feels much lighter and weaker and doesn’t move the same. I want to be a part of the remainder of her life. I don’t want her to pass away thinking that my neighbours are the ones that loved her the most and that I abandoned her. I feel so guilty for everything and genuinely want her to be happy in this house and be like how it was before.

I have considered some options. One is that I could keep mango in one room for a couple of weeks and keep Tj inside. This way she can roam around the house without coming into contact with Mango. And then I can start the reintroduction process again but do it properly this time. I still need to try this but I just want to see if there is anything else I could do so any advice would be much appreciated.

Ultimately, i just want both my cats to be happy and healthy and a part of my life. Tj is a priority because of her age and I really want her to know how much she means to me. Any opinions or advice would be really appreciated. I am willing to try anything at this point. Thank you for your help in advance!!