All you need is a can of Minwax lacquer, gloss, black rattle can, and spray on days under 50% humidity (and don't leave out overnight). Done it a million times. You can get into blush eraser if you're rushing and trying to spray on higher humidity days but that's more cost and more potential problems.
So despite wanting satin I'm suggesting the gloss lacquer. Why? Because you can polish it to a satin point, and without unnecessary solids added like in the semigloss and satin varieties. More or less, wet sand/sand to the point you want and stop, you're not going to "accidentally" get a showroom high-buffed shine by accident. I've found this stuff for as little as $6 in Lowes but that was a while ago. Amazon usually has in stock
https://www.amazon.com/Minwax-15200-Brushing-Lacquer-12-25-Ounce/dp/B0002YOPOG/ref=asc_df_B0002YOPOG/?tag=hyprod-20&linkCode=df0&hvadid=193184886406&hvpos=&hvnetw=g&hvrand=11420564240574443059&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9007455&hvtargid=pla-332269640634&psc=1
I go about 2 coats/day 8 total (each coat a couple passes). First coat nice and easy. That's enough for you to gently sand level if you're not heavy handed but I would start no higher than 800 grit.
Do NOT spray in the house. There are tons of other options low VOC if you want but lacquer, nope. Not even a little for a headstock, don't ask how I know I just know