Hi Pete, the current Fusion Worlds mode is conceived as an Arcade mode. So in order to finish the game and unlock the next difficulty you have to beat the game from the beginning. The game already save your progress between levels in case you quit the game to be able to continue it later, but, as you said, if you run out of lives your progress is lost.
We are working really hard on the new Skills system that adds a lot of variety in each run and will help a lot to finish the game in the normal difficulty. We are also redesigning the difficulty levels to change completely the experience in each level making them a new challenge.