I was thinking something similar. I vaguely remember that the characteristic function proof includes an assumption of n being large, where n is the number of variables being summed. I think that allows you to ignore some higher order n terms. So by keeping those in you could probably get some way to quantify how “close” a resulting distribution is to Gaussian. And you could relate that back to moments quite naturally as well.
I was thinking something similar. I vaguely remember that the characteristic function proof includes an assumption of n being large, where n is the number of variables being summed. I think that allows you to ignore some higher order n terms. So by keeping those in you could probably get some way to quantify how “close” a resulting distribution is to Gaussian. And you could relate that back to moments quite naturally as well.