I think that your hair growth is too coarse for removal by depilatories.
It appears that at least some of the hair in your moustache is male-type hair and that's going to be tough to remove by any method.
Laser isn't at it's most effective other than on black hair against a light skin. Both laser and electrolysis depend on getting your hormones into the normal "female" range before the removal is effective long-term.
Waxing becomes less and less effective the tougher the hair growth is.
If you're shaving it might be a good idea to try one of the Philps Norelco razors from the men's range of rotary-headed razors.