That is great that you have such a huge heart towards those not included. Over the years, I have found some things that have helped me with these primary age students.
I don't allow students to choose their partners in the classroom ever. When I do, I can always see the pain of the ones without partners. Even worse it interrupts instruction. I use to allow choosing partners, but after about 15 years of teaching, I said to myself, "No more". Instead, I often randomly choose partners such as with playing cards. I have 2 of each card and just enough cards as students in the room (i.e. 24 students leave out the Kings and have 2 of each card. That equals 24 cards. Each child gets a card (or picks a card) and the 2s match up, the 3s match up etc. To save time, I might only switch partners once a week.
In your situation with your challenges, I might get a bit more creative. This takes time, but it can be the solution to your problem long term. Have each child write down the 4 students they would like to work with. With primary, I would often make copies of a class list, and give one to each student. They must first put their name on it. Then, each child puts a 1 by their first choice, 2 by their second, etc. This must be really private or it will backfire on you big time. Then during your prep time or after school, find out if there are any students that chose one of your not included students even if it is 4th. Then play match maker and secretly pair them up. I have done this and some students make their first friends by doing this.
Lastly, observe carefully the students who are not being included. Observe to find the top top 2 reasons students seem to not like working with them. Then maybe some boundaries might need to be set up. For example, if they always are talking, possibly having a rule that each time one talks, they have a colored chip. Once they put in their colored chip, they can't talk again until the other student puts in their colored chip or a variation of this.
You may have to tweak these a bit for your students and based on your knowledge of your students. Hope this helps some.