Most "kids cameras" sold today just use cheap webcam sensors (e.g. 1 MP, low dynamic range) that are sold for excessively high prices. They have few physical controls, no viewfinder, and are bulky.

Instead, why not grab a used iPhone SE, the camera sensor is still fantastic, and it will likely cost you less than most kids cameras. Remove everything except the Camera App, leave it in Airplane mode, and it will last roughly two days on a single charge (over a week idle).

PS - You can find deals on used cellphones by looking for "network locked" ones, since you won't be putting a SIM in it anyway.