K.P.
Maybe call the mom and suggest taking the girls out for ice cream or a bite to eat after class and go from there! The other mom is probably thinking/feeling the same way, so just mention that the girls want to hang out together outside of class, do they want to grab a burger next week.
If the mom says no- there's your answer. If the mom says "yes", then you have a chance to assess eachother over Diet Cokes and fries WHILE the girls get a playdate (supervised). If it all seems okay... then you don't need to have a mom-date again unless you want to!