Have you tried the child's farm brand? A friend of mines child had a reaction to nivea or one of the popular brands and our nursery recommended Childs farm. It may work for both you and the baby. In terms of application we use this roll on applicator that you can use whatever sun cream you like in it. It had foam edges to sort of rub it in so again might work for you. It's made by solar buddies. https://amzn.eu/d/7rc7vro
I use a stick. As they become toddlers, they can apply it themselves fairly early on. I'm not allergic to sunscreen, but chasing down and forcibly applying a lotion to two screaming toddler faces is enough to give me hives...