but my son’s bike doesn’t have a place to attach anything.
There is always a way to attach a basket, it might not accept the same type of basket you husband has but there are companies that specialize in attachment hardware for hard to fit bikes.
put his hat in a shopping bag and hang it on the handles,
Not a good idea. Loose weight hanging from the handlebars can make a bike difficult to control, also it can get pulled into the front wheel.