Vestigial Filter problem?
Your reasoning seems reasonable and correct to me. Assuming that the filter's impulse response $h(t)$ is real, then its magnitude response $\left|H(\omega)\right|$ will be symmetric about $\omega=0$, so the negative-frequency terms will be scaled by the same values as those on the positive side of the frequency axis. I don't see any reason why you would need to multiply or divide by 2 anywhere.

The only thing that might differ if you see a solution somewhere else might be the scaling factor at the front of the expression. The typical transform using angular frequency would be:

$$
\mathcal{F}\{\cos(ax)\} = \pi\left(\delta(\omega - a) + \delta(\omega + a)\right)
$$

which assumes a Fourier transform definition of:

$$
\mathcal{F}\{x(t)\} = X(\omega) = \int_{-\infty}^{\infty}x(t) e^{-j\omega t}dt
$$

It looks like you have an extra factor of $0.5$ at the front of your expression that you might have gotten from the transform. Note that different implementations or expressions of Fourier transforms often have arbitrary scaling factors included; what's most important is that if you apply the forward and inverse transforms in sequence, you get the original signal back:

$$
\mathcal{F}^{-1}\left\{\mathcal{F}\{x(t)\}\right\} = \mathcal{F}^{-1}\left\{X(\omega)\right\} = x(t)
$$

or at least to some known scaling factor.