FFT of BPSK signal interpretation
A random sequence of +1 and -1 is not really a BPSK signal; it's just a sequence of bits. As such, its FFT has not much meaning, at least not in the sense you're thinking of.

To create an actual BPSK signal, first denote your sequence of $N-1$ bits by $a_k$, where $k=0,\ldots,N-1$ is an integer. Second, choose a bit rate $R_p=1/T_p$. Then, choose a pulse shape, $p(t)$. The pulse shape is typically rectangular, a sinc, a raised cosine, or similar. The important thing is that the pulse is orthogonal to $T_p$-shifted versions of itself; that is, $$\int_{-\infty}^\infty p(t-mT_p)p(t-nT_p)=0$$ for all integers $m\neq n$. Now you're ready to create your BPSK signal, which is given by $$s(t)=\sum_{k=0}^{N-1} a_kp(t-kT_p).$$

Now you can find the FT of $s(t)$ (or the DFT of a properly sampled version of $s(t)$), and you can find the bandwidth and other spectral properties of your BPSK signal.