Wood – Non-Toxic Wood Glue

This-to-That recommends JB-weld or LePage's Metal Epoxy. I've never used either, personally.

The important thing for this is going to be surface prep and clamping, I would assume.

You might be able to use a CA glue (ie, "super glue"), but you'll need additional reinforcement to keep the bond from shearing (eg, glue then bolt it in place, but 1/4" is kinda narrow to bolt through; also, I have no idea what sort of clearance issues there might be if there are bolts sticking out)

Use a hard, inexpensive local wood, without large pores. Hard, since they will get banged around. Pores will allow the wood to trap bacteria, dirt where you don't want it.

Maple is my first choice. Second choice might be birch, sycamore, or alder, if you have a source for them. Poplar is not bad either - cheap, reasonably tough. Cherry would work too. Much depends on what you can find locally. Do NOT go to your local big box store (Home Depot, etc.) to buy hardwood. Hardwood bought there will be quite expensive for what you get.

Woods that I would NOT use are imported exotic woods. These can often cause allergic reactions. Another wood to be avoided for this purpose is walnut, which might cause reactions in some.

Use wood coloring, rather than paint. Paint can chip off, and may not be food safe. And be careful with stains, since many of them will not be food safe. Wood dyes, like aniline dyes, may also be something I would not put in my child's mouth. For a coloring, I would use simple food coloring.