I participated in a sleep study but I was an adult. I think that you'd be surprised how much people sleep in the lab - I didn't think I would either, and the technician told me that everyone feels that way. But because we are all sleep deprived, we DO fall asleep. They don't expect him to sleep well there when he doesn't sleep well at home - they want to know for sure how well he ISN'T sleeping. The advantage is that the electrodes and read-outs will tell you/them WHY he's not sleeping, and what's going on at the time he wakes up. Obviously your three year old will not be able to do the survey that adults did - but when they asked me the next morning how many times I thought I woke up, I said 3. Two on my own and 1 with the technician in there adjusting a wire. Guess what? I actually woke up 130 times!!! I definitely had a sleep issue! It had nothing to do with being in a strange place - it was neurological. I learned a lot, and a treatment plan was decided upon. Being without REM sleep is awful, and it's not safe. I should not have been behind the wheel of a car, but I didn't know it. Lack of sleep is serious.
If you go forward with the study, you can make a game out of the electrodes with your son - beforehand, read some books about astronauts or robots, then say you are pretending to be a robot that night. R2D2 or whatever piques his curiousity! Nothing hurts, and it's very interesting getting all hooked up. I assume they will allow you or your husband to sleep overnight in the same room - my room had an extra bed in it for that purpose. I'm sure the technicians have special things they do for kids to make it fun and not scary at all. My room had a bathroom and a shower, so maybe you can tell him it's a hotel or something else fun. I'm sure you can take some of his things too - stuffed animal, special blanket, maybe a CD player, books - whatever his bedtime routine is, and whatever comforts him and makes him feel "at home"! I don't know what the room will look like where you are, but my set-up was very non-medical looking. The electrodes were stuck onto my head, with one or two on my chese and one on my leg. Everything connecting to a small device that looked a lot like a TV remote. All of the other equipment is elsewhere, in another room, out of view.
Above the bed is a darkened window in the ceiling, kind of like those cars with dark windows - you can't see in but you can see out. There was a camera up above that allowed the staff to monitor my movements, but I couldn't see anything. They didn't turn the camera on until I told them I was going to bed, so it's not like anyone was watching me get undressed or anything. I don't think they actually watched me all night but I think they tape it. Maybe they look in now and then especially if they get a reading from an electrode that indicates that something is off or disconnected - I don't know but you could ask. That was it for the room - it was kind of sparse looking but there was nothing scary looking - not hospital-y at all. Maybe you can get a quick tour of the sleep center beforehand just to answer your questions and calm your fears, and that will transfer as calm to your child.
Definitely get good info from the doctor - if you're not comfortable, or even if you are - get an opinion from a pediatric neurologist. Don't worry about insulting your pediatrician by asking for a consultation - they are NOT insulted in general, and if they are, it's time for a new pediatrician. The results of the tests must absolutely be interpreted by a pedi neuro.
Since it's only been going on for a month, there's probably no immediate threat of damage being done, so I would think you would have time to make an appointment with a pedi neuro for a pre-sleep-study consultation if you want one. Then you can decide on the sleep study.
I think the hyperactivity connection is something they need to evaluate too. One could be causing the other, but right now it's "chicken and the egg" - hard to know which one comes first. You could also ask what happens if he goes to bed at 7:30 and wakes up at 3, but then can't get back to sleep with all the equipment - they may tell you that they have enough info by then on what was going on.