The entire cup would not have to be heavier. Only the bottom. If you were to use a material like cork or even just increase the diameter of your bottom so the center of gravity is closer to the bottom you could make it less likely to accidentally tip over from a bump. If you decrease the height and increase the diameter you can make it sturdier while keeping the same volume / keeping prices down.
Hi, this cup looks like a great idea. From the little time I've spent with it, I think it would be a great idea to use a textured pattern on your cup to increase the grip (This can be done by heating paper and using a rubber stamp). Another suggestion is to remove the side gaps from the cup to improve its resistance to spills. This can probably be done by folding down the sides or even using heat /food safe silicone. You could also add in a suction cup or use a heavier material in the bottom so that your product is more resistant to accidental pushes.