Here’s what I’m thinking, let me know if you think this approach would work. This approach would assume the earring would already be in the ear.
Basic idea is to use tips that fit on an existing device, scissors for example, making the solution quicker, cheaper and perhaps customizable over designing the full device from scratch.
Design two tips to fit on a pair of scissors. One tip would be a flat or slightly concave surface to press against the earring. The other would have a small “tray” to hold the back. The tray would have a countersunk opening to guide the post into the hole on the back. You’d place the flat tip on the earring, and close the scissors to place the back on the post.
Scissors close in an “scissoring” motion rather than an arc. The tips would have to factor this in. Something more like C Clamp locking pliers would have a better motion but not as readily available in a small size. There may be other, more suitable, options.
Safety: Scissor blades would need to be long enough so that there is no chance of users getting cut
Damage to the earring: Flat surface would have to be soft enough to avoid damaging the earring
Variety of earrings: Not sure a single surface would be good for all earring types. A variety of tips could be designed to be interchangeable.
let me know what you think.