That is the first mistake in your train of thought, the universe doesn't expand "at the speed of light", it is not a ball expanding outwards with an outer shell expanding at a measurable speed.
Instead it expands everywhere all at once, all distances are being stretched. And this streatching doesn't have a "speed" (m/s), there is nothing traveling, so it is not bound by the speed of light. Instead it has a rate of expansion (m/m/s) as in how much each meter will grow in a second.
Sorry, I'm not an expert so I don't know how to explain properly.