How do I stop padauk dye bleeding into my oak lid?

Seal the padauk first, and yes, shellac is the usual move for exactly this problem. Orange dye bleeding into whatever it touches is just what padauk does, and it'll keep doing it for a long time even after the wood looks dry. I did a box similar to yours a couple years back and learned this the hard way on a rag I left sitting on the base overnight. Pink stripe across a maple offcut, permanent lesson.

A couple thin coats of dewaxed shellac on the base only, let it dry, sand it back lightly, then you can finish the oak however you like without worrying. Shellac does mute the color a bit while it's wet, but it comes back as it dries, and once your topcoat goes over everything it looks the same as it whould otherwise. If anything the shellac makes the red look deeper and richer rather than duller. Use the dewaxed kind if you plan to put poly over it, the wax in the standard stuff can mess with adhesion. Skip the Danish oil on this one. Oil finishes are thin and they'll pull that pigment around, expecially anywhere two woods meet. Wipe-on poly is fine as your topcoat since it's a film finish, it'll sit on top and lock the color down instead of wicking it sideways.

For the joint line, the trick is finish the lid and the base completly separately while they're apart. Finish the recess in the lid, finish the base all the way into where the lid sits, don't assemble them til everything's fully cured. If you finish it assembled, the orange dust and any bleed wicks right into the oak at that seam and there's no fixing it after. I also blow off or vacuum the joint before final assembly, padauk dust has a way of hiding in a fitted recess. One more thing, wear gloves and a mask while sanding that padauk. The dust gets everywhere and plenty of people react to it.

Same dust headache here a while back turning a padauk bowl next to some cherry parts on the bench. The thing that got me wasn't the finish, it was the sandpaper. The grit loads up with that orange powder and then the next piece you touch with it picks up a pink haze. I keep a seperate stack of abrasives just for the red woods now, problem solved. Cheap fix after learning that one the hard way. One thing to know going in, that bright red will slowly shift toward brown over the years if the piece lives in daylight. Nothing wrong with it, some folks like the aged look, but if the red is the whole point of your design maybe keep it off a sunny windowsill.

Gonna push back on the idea that shellac makes padauk look deeper and richer. In my expirience that depends a lot on wich shellac you use. I did a padauk mudda... a padauk handled spoon rack a while back, and the amber stuff pulled the color noticeably toward orange-brown. It never fully snapped back after curing either. A clear or super blonde cut fixed it, but I'd still say test a coat on a scrap before you commit to the base. And while you're at it, leave a test piece in a sunny window for a few days. Padauk darkens on its own over time anyway, so what looks right fresh off the brush might shift on you.

Update: shellac on the base worked, and the red still looks bright through it. Also switched to a dedicated pile of sandpaper for the padauk after reading about the grit picking up the dust, that tip alone saved my lid. Still nervous about the joint line though. If finish gets in that seam, will the lid end up stuck in the recess over time?