What finish is actually safe for a toddler bed made from mystery scrap wood?

The unknown species thing matters less than you'd think. What actually matters is whether the wood was ever pressure treated or coated with something nasty, and there are easy ways to check. Pressure treated lumber has that greenish tint and you'll see little incision marks on the face where the chemical got injected. Give the boards a sniff after sanding a test spot, if it smells like chemicals or copper rather than plain wood, don't use it anywhere near a sleeping kid. Same goes for anything with old paint, stain, or glue residue. You can rip those faces off if the board is thick enought, or just set them aside for something else. Mill stamps are fine to ignore, by the way. Those just tell you the grading agency and mill, they don't tell you anything about treatment.

For finish, since she chews on everything, I'd skip anything with a warning label on the can regardless of what the internet says about curing. Pure tung oil is food safe and cheap, or there are hardwax oils marketed for kids toys and cutting boards. Both cure hard in a few days. Give it a full week before she sleeps in it just to be safe, longer in a cool room. Wipe on, wipe off the excess after ten minutes or it gets gummy.

Before finishing, sand everything to 150 or 180 grit and vacuum it well. That does double duty, it cleans off any surface contamination and it blunts all the corners. Little kids climb and flop around on beds, so hit every edge and corner with a roundover or even just a few passes with a sanding block. Sharp 90 degree edges on a toddler bed are shin magnets.

One more thing since you went with a solid plank instead of slats, drill some holes or cut a gap in that bottom panel so air can move under the mattress. A sealed box underneath is how you get mildew and a musty smelling mattress within a year.

One thing I'd add since it's job site scrap, check every board with a magnet or metal detector before you run a plane or sander over it. I built a bench out of demo leftovers and found a staple buried an inch deep that took a nice chunk out of my blade. Also if any of it had old paint on it, those lead test swabs are a few bucks at the hardware store and worth using on a kid's bed. Better safe than sorry there.

One thing I'd flag about the mixed species part is that whatever finish you pick, it's going to look different on every board, since pine drinks oil while something dense like maple or oak barely lets it in. I made a blanket chest out of a similar grab bag pile and the first coat looked blotchy as anything til I realized the contrast was just the wood behaving differently. Now I always test on offcuts from the same pile first, and I'll add that after the first coat of oil the grain will stand up and feel rough, so a quick light scuff with a fine abrasive pad or paper between coats gets you a much smoother surface, wich matters extra when you've got a little one running her hands over every inch of it.