This is a very common problem. Most people expect the older cat to keep the youngster in line but frequently it is the new young cat that causes the resident cat to move out. It's happened to me as well but luckily my cat didn't find a new home & eventually he got used to te new kitten & they became quite good friends.
I don't know how you made the initial introductions but I think that you need to go back to the start as Beans is probably feeling very unsecure & unhappy that his previously safe home has been invaded by the "kitten from hell"

. I know that Toffee is not really the kitten from hell but this is how Beans will see it. Unless you can convince Beans that his home is safe, secure & that he has his very own space that Toffee can't get into, Beans may decide to move out completely.
Firstly make sure that your neighbours do not feed Beans & try not to let him into their home.
Next, can you set aside a room in your house that is exclusively for Beans & that Toffee isn't allowed into? In this room you need to put everything that Beans needs, i.e. food, water, scratching post, toys, litter tray if he uses one, & most importantly, a variety of warm beds, preferably off the floor.
Put Beans in this room & shut the door for a while. Make sure that all members of your house visit Beans in his new comfort zone, spend time cuddling him, grooming him, playing with him, whatever he likes to do & anything that makes him purr.
Try to ensure that Toffee doesn't go in there but make sure that when Beans wants to go out, Toffee can't ambush him, attack him or even see Beans, if possible. The route into & out of Beans' new safe area must be a "Toffee-free zone" as much as possible.
Beans will need to spend some time chilling out in his new room & destressing & this may take a few days. During this time you need to get Beans & Toffee used to each others' smell but in a way that provides positive associations for both cats. So when any of you familly have been cuddling or playing with Toffee & are covered in Toffee's smell, then go into Beans' room with some extremely tasty treats that he only gets at this time & try to hand feed him.
After a few days, try to hand feed him with the treats when Toffee is outside the room (but with the door shut). Beans will know Toffee is there by the smell. Then gradually open the door a few inches & get someone else to distract Toffee with treats or toys while you hand feed Beans.
Once you can do this with the door open but both cats distracted & ignoring each other, then you are well on the way to allowing them to meet in other parts of the house.
Make sure that all meetings are positive experiences for both cats, i.e. the cats ignore each other while you & your family distract them.
If you can get to the stage when both cats tolerate each other, without any signs of aggression, fear & anxiety, this will be a huge bonus. Latest research shows that it can take up to 2 years for strange cats to become friendly & tolerant towards each other so don't rush things & don't be disappointed if they don't become bosom buddies. If they can learn to share your house amicably, this will probably be the best you can hope for.
Good luck with them both ~ I hope it works out for them.