Goodness gracious rent a car. I cannot believe all this talk of Metrolink. You're going to drop way more coin and time messing around with cabs and Metrolink for your crowd then you will on a two day car rental which is dirt cheap at STL. The bus to Pappy's? Is that a joke? Honestly...
There is endless parking all over the place even at WashU. (my alma mater as well).
With the jumping around you might want to do; Washu, zoo, Ted Drewes, etc. the car is going to suit you much better. Also, you really want to drive your kid around and let them see what that area is all about. Clayton has some amazing neighborhoods with classic houses and mansions that just need to be seen. Drive that kid around Forest Park to at least go by the Art Museum, Muny, Science Center, etc. Drive along Euclid and Maryland in the Central West End for a taste of that.