Smile through gritted teeth is an idiom which basically means you are doing something you have accepted to do but don't necessarily agree with it. For example, if you hate going to the gym but know it is good for you, you will grit your teeth and go through with it. That might not be the best explanation, it is hard to explain! I hope this helps :)

To rasp through gritted teeth means that you pretend to be happy or supportive while speaking an unsupportive message. To smile through gritted teeth practically means the same thing. Sorry if the answer is a tad unclear, it's quite a difficult thing to describe.