Yes. I think a game like this can either choose to reset the player's progress upon death to the start of the level or work with a timelimit and deduct time whenever you are hit as you propose. Currently it feels frustrating when you die at the end of a level, since not only do you need to redo the level, but you also won't be able to get very far into the game since you lost so much time.
In general I dislike timelimits in games, especially in this one. It runs counter to the story elements you have in your game. The feeling that I'll have to play optimally to see all the content really makes me not want to put effort into it. But that's a personal gripe of course. I guess what I'd like the most is for the timelimit to be optional, something like a high score type of thing. But your suggestion would alleviate a lot of the frustrations.