Do I need to bandpass filter the signal before applying the Zoom FFT algorithm?
Your BPF is superfluous. Frequency translation (if very simply done by $\cdot e^{j2\pi \Delta f n}$) is a circular shift of the whole Nyquist band. 

When you calculate the Fourier of that, you'll notice there's no chance of aliasing at all.

The only step that leads to aliasing is the downsampling, and that's why ever decimator comes with an appropriate filter - in your case, the LPF you already have. You usually implement the filtering and down sampling as one entity - no use in calculating filter output that you'll throw away in the next step.

So: 

$\cdot e^{j2\pi \Delta f n} \rightarrow \text {decimating LPF}\rightarrow \text {DFT}$