I would use a set of cabinet scrapers.
There is a little extra setup because you need a vice of some type, burnisher (very hard steel rod), and a bastard mill file (and/or some hard sharpening stone like DMT diamond impregnated stones, I got mine at home depot for $30).
You can find lots of videos on how to sharpen these things, but it comes down to making a square edge at the end, work hardening it by rubbing the burnisher on both sides next to the edge, then putting the scraper straight up and burnishing the top and then tilting the burnisher to "turn the burr" which makes the end like a little hook.
Then you push or pull this edge along and it will shave off very thin (type II-III) wood or poly shavings.
It sounds like it is more trouble than it is worth. But for a project like this it will very quickly take off even shavings until you get down to even wood. Sandpaper, in my opinion, is going to lead to more wobbly edges as it is kind of difficult to sand perfectly evenly, especially around corners like that.
On top of that a single scraper will last forever, and has all but replaced sandpaper for me (except for some very deep bowl shapes). They also come in all different profiles to match different edge profiles.