Why can't one use the Cook-Levin theorem to show that TQBF is PSPACE-complete?
If you want to write $N$ words of output, it takes at least $N$ steps of computation to do that.  So, if you need to produce an exponential-sized output, you'll need to spend exponential time to do it.  For that reason, a polynomial-time reduction cannot produce an exponential-sized output when run on a polynomial-sized input.