Separation of sinusoidal oscillations from the frequency spectrum
You don't have to perform a full FFT. Take the definition of DFT:
$$X[k] = \sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{N}kn}$$
What you need is one element of the sequence $X[k]$. For example, suppose the sampling rate is $f_s$, and the signal has a length of $N$, so the frequency resolution is $\Delta f=f_s/N$, and $k$ corresponding to 1000 Hz equals to $1000/\Delta f + 1$. Just do DFT at that discrete frequency $k$.
On the other hand, if you want to convolve your signal with something, use sinusoidal wave of 1000 Hz. We know that a convolution in the time domain equals a multiplication in the frequency domain. The Fourier transform of sine wave is delta function at the specific frequency. So convolving with sine wave can extract the harmonic component.