Mom of teenager here. Been there.
If you do a play date for a group, as noted below -- always even numbers. All four girls, or just two, never three. And I'd add: Have some specific activities ready for them to do, especially the first time, rather than telling them all, "OK, now go and play" and leaving it all up to them. Plan a craft and be around to encourage that they do it. Or take them somewhere. Be ready to suggest things if they start to carp at each other. That's all IF you do a four-way play date....
Because frankly I wouldn't.
If they already are having issues playing together at school, tossing them together into a four-girl play date at home isn't necessarily going to go any better; the same "You're my new friend, don't pay attention to your old friend" dynamic will still exist, even though they're at a home and not on the playground. They might be less assertive and surprisingly subtle about the rivalries because moms are nearby, but the issues will still be there. It's asking a lot of kids this age to "get to know each other"; they won't want to be guided by the adults, and probably feel they already do know each other. You might find that the girls themselves will tell their moms they don't want this play date if they realize in advance that certain other girls will be there. And surprising them with "Look, Sally (your rival) is here too" as they walk in the door -- well, at this age, that kind of surprise might backfire. I think a four-way play date might be too much effort at trying to manage this for them, though the intention is good.
That's why I'd probably do separate play dates for your daughter with Suzie, so they can keep connected, and with New Friend at other times. You are already doing a great job with talking about how friends come and go, and I'm sure you're talking with your daughter about how she should not let anyone else tell her who is her friend -- it is NOT bullying, at this point, but can veer in that direction if the other girl is very possessive. I'd role-play with your daughter about how to say to New Friend, "We've played kickball and now I'm going over to see what Suzie is doing" and how to handle it if New Friend says "You can't!" and especially if New Friend gives her the cold shoulder the rest of the day.
One thing about play dates with New Friend -- If the girl goes to after school care, you can talk with her parents about picking her up from after school care to go home with you. People used to do that all the time at my daughter's elementary school for play dates. You and the mom both will need to find out what the after school care staff requires in these cases -- likely you'll have to have the other parent inform the staff, possibly in writing, and you might have to present an ID when you pick up the other girl. That's how it worked in our public school after-school care. It's not as big a deal as it sounds and the girl might like the break from after-school care.
You mention after-school activities and that's great, but be sure to have your daughter in something that isn't school-related too. If she is at after-school activities and these same girls, even Best Friend and New Friend, are in them, the issues are still hanging in the air. It's good for kids to have something of their own that is not with the school crowd -- Girl Scouts (shoot for a troop that isn't based at the school if you can; not all are and no one is required to attend a troop that meets at her own school), or dance or a sport etc. where it's not a gang from the same school. Look for things where the kids participating are from a variety of schools, too, so it's not all a gang from another school and then your daughter solo.