Should I consider systems to be causal or non-causal by default in questions like these?
Are you confusing the Fourier and Laplace transforms? The Laplace transform is normally defined as one-sided [1], but can be defined as two-sided. The Fourier transform is two sided (as far as I'm aware).

One way to solve this might be using Parseval's theorem,
$$
\int_{-\infty}^\infty | v(t) |^2 \, \mathrm{d}t   =  \frac{1}{2\pi} \int_{-\infty}^\infty | V(\omega) |^2 \, \mathrm{d}\omega,
$$
and changing the limits of integration from \$\infty\$ to \$a\$.

[1] https://en.wikipedia.org/wiki/Laplace_transform#Formal_definition