3 year old having troubles sleeping

Based on what you described, my thought would be that supporting emotional release (particularly during the day) would be a way to help him sleep better. It makes sense that he is touchy & whiny during the day, probably both because he isn't well-rested, and because he is carrying around a load of feelings that drive him off-track (and then especially interfere at bedtime, when things are quiet & he is trying to settle.)
How do you typically handle his "touchiness" during the day? Is he rigid, making demands that escalate or get increasingly unreasonable, but seeming unsatisfied with your attempts to accommodate him? Is it like the rest of you are walking on eggshells, and the slightest thing can bring on tears or tantrums?
This kind of pattern signals that he can't think well; he's too emotionally charged to function well, to be flexible & resilient. A combination of setting a reasonable limit kindly AND staying close when he

gets upset in response to the limit, gives the support that lets this emotional release perform a healing, restoring function. This doesn't happen when you focus on avoiding upsets (making things go his way so that he doesn't fall apart or melt down) or when you respond negatively to his emotional outbursts (ignoring them, punishing them, sending him to his room "until he can talk nicely," etc.) If you can offer connection to him & know that what he needs is to get out the feelings that are interfering with his good thinking/good nature, then you can support him when he is offloading the stuff that is causing problems for him.
This can also come up at night (oftentimes, it's unresolved feelings about separation) as well, and your focus would be to support him & facilitate his release with your caring attention (not soothing him out of his upset, though), but it's probably easier for everybody if you do the work in the daytime.