L.R.
I think telling your daughter that she should end the friendship with Sophie, as someone suggested, would turn Sophie into "forbidden fruit" in your daughter's eyes. Chloe will suddenly think of how much she really does like Sophie and will resist your attempt to end the friendship. I would not give Chloe any lectures about dropping a manipulative friend -- especially as you say this girl really is OK and a pleasure most of the time! She is being five and likely she gets her way with tears and fussing at home, and thinks she can do the same with your daughter. It is typical for some kids. If you and Chloe like her otherwise, why drop her over this? They're not middle schoolers in a mean-girl situation; they're five and still barely able to navigate a world with other kids in it.
After what you described of Sophie's ways, I would not have walked Chloe over to their house "to talk about it." If Sophie knows you came over, it fed her drama. I would have left it alone, but it's done now. Next time just acknowledge it and then distract your daughter: "It's too bad Sophie feels that way. She really wanted a yes to a playdate and she got a no. But sometimes we all have to hear no and be OK with it because we have to do other things. Now, how about a snack and getting on with that homework? What's the homework today?" And so on. Your own daughter needs to hear that "No" will happen -- and you will stick to it-- no matter what fits get thrown by anyone involved.
Next week, invite Sophie over and never mention today again. I would cut back all the after-school play dates not just with her but overall. And if you are friends with the mom, you can mention that Chloe was worried that Sophie was upset. I would be the mom is well aware of her daughter's tendencies to have a "bad list" and to cry when told no.
If this keeps happening to the point of these teary fights then you need to find some other things for your daughter to be doing and limit their time together, especially if they are also together a lot in class. And to answer your final questions -- at five they still need help "working things out themselves." If they get into it in front of you, it's time to intervene with "Sophie, we don't have a 'bad list' in our house. Everyone's friendly if they want to play together, or they go home until they feel like being friendly to everyone" or whatever the situation calls for.