How can I get rid of this unexpected minus sign on my inverse Fourier transform of two impulse functions?
HINT: you should review the sifting property of the Dirac delta impulse:
$$\int_{-\infty}^{\infty}f(t)\delta(t-t_0)dt=f(t_0)\tag{1}$$
if $f(t)$ is continuous at $t=t_0$.