Until she gets used to using a big potty on her own, I would just buy some type of tote bag that the Blue's Clues seat would fit it so that you can take it everywhere. If that's what she uses at home & all the places she feels comfortable, then it's understandable that she doesn't want to use a big public restroom potty (heck, I hate using them!!).
Then, once she is totally consistent with using the potty every time, then home would be the best place to start introducing the big potty alone. When she gets comfortable with using no potty seats at home, she will be more likely to ditch the seat in public places.
IN THE MEAN TIME, here is something you might consider (see the link below). It is a web page with several more portable solutions...such as smaller, fold-up, travel potty seats. There is even a Blue's Clues one. They are much easier to take with you. You would probably need to get her used to using it at home first, but it might be a way to ditch the bulky seat all-together.
http://www.pottytrainingconcepts.com/Travel-Potty-Seat-c....
GOOD LUCK!!!
C.