Hi helios,
i'm a novice in FFT. In fact I am taking a DSP course which discusses the FFT algorithms. I haven't implemented the FFT alg. in any processor like you are doing now. But theoretically, when you use FFT alg. then won't use the same number of multiplications as the original DFT formula, i.e. the twiddle factors should be much less than 2048, because this is what FFT alg. does to reduce the computational burden.
Take for example radix-4 FFT algorithm. In this case you just need 4 twiddle factors (numbers) for all FFT butterlies in all stages, plus some additional factors. In this way 1K memory space is more than enough.
best