Some sort of planer or scraper is probably what you want. Sanding can take edges off bumps, but isn't actually that great for getting things flat. (Well, not unless you put sandpaper on something very hard and flat. e.g., when truing up metal surfaces by lapping them on fine grit paper attached to a nice flat surface.)
Since you are going to the effort of creating a nice piece of cabinetry, it may make sense to use a cabinetry finishing tool, and take off targeted curls of material where the bumps are, letting your hand and eye tell you when you've trued things up enough. As per the comment, use a straightedge to get an idea of where the high spots are. Mark these and take your time to get the panel down to reasonably true.
I actually wouldn't use a belt sander for this purpose, as it seems a bit overkill. Belt sanders, to me, are for taking off material fast, but as a finishing step I think I'd be afraid it was too aggressive. Not to mention that I think the main issue you are going to have is that the material you are trying to level is a mixture of hardnesses, so the softer material will always come off faster. Sometimes you end up chasing trueness as you gouge out the softer material faster.
Luthiers would use calipers to test thickness, making pencil marks where material should come off. Then they scrape, test, mark, repeat. You can do this, too, if you want a very nice true finish. This might be overkill for the purpose discussed here, but if you have marking calipers these can work similarly to a straightedge.
There is an art to using and maintaining scrapers for this purpose, but the results are way better than sandpaper (IMHO). Once you get the hang of it, it is super easy to target fine areas without touching areas around it -- something sanders are just not that great at.
You can find more details about scrapers by searching "finishing scraper" in your favourite web search. That got me enough results to fall down a rabbit hole of related information!