I have a large mono Bucket, vernis Bucket GM, a Sac Shopping and a Cabas Alto. None of my bags have tipped over in the car and let me tell you I DRIVE and CORNER. If they do tip over everything is inside of it's own compartment.
The vernis Bucket is very, very stiff. It doesn't give at all.
The way to keep your things in a convertible bag, in my opinion, is to compartmentalize everything. I don't have anything loose in my bag. Being an accessories hog, I have a LV compartment/pouch/pochette for everything. The things that would be considered loose are still their own holder ie. Agenda, Mono Pen holder, Mono Sunglass case (that I don't use), mono coin purse... You know what I mean? The key is to compartmentalize. Buy LV or someone elses accessory holders/cases then pack your back and sport it!
Also by placing things in various compartments you can always reach in and get what you're looking for and one doesn't have to worry about someone reaching inside of the bag and grabbing some miscellaneous thing. The pouches/pochettes/holders stack down lower in the bag.
Lastly, if you're in the car alone you can always strap it in with the seatbelt!