But seriously being the parent of the emotional kid can be draining, I've been known to use "oh just suck it up" on some bad days!
I think if your uncomfortable with your friend disciplining your child you should say so openly.
With us, the friend we no longer see was because it become to hard, we were very different parenting styles, as she looked at it as her way was right and mine wrong- so after kids our friendship was doomed.
With other friends what I found was best was not to intervene so much, let them work it out. DD was never one to fight for a toy, she would cry or move on. We tried to equip her with ways to manage situations.
At that age she'd firmly say "stop- that hurt my feelings" it was something one of the cc carers had her do, it sounds silly but really worked well a lot of the time. She didn't feel forced into something and the other kid would be like ok then lets do something else and they'd quickly move on.
Sometimes we get overly involved in these issues and complicate them for the kids, especially when we are the parent of the emotional kid, in the long run they really need to learn to fight their own battles.
Hope your able to work it out with your friend!