Hashing new user's password (extending user-permissions auth.register)

Have you tried the lifecycle hook?

i believe the right choice for your use-case is to use the lifecycle hook beforeCreate

import { createHash } from 'node:crypto'

module.exports = {
    async beforeCreate(obj) {
        let { params } = obj // attributes model, state

        const password = obj.data.PASSWORD_FIELD // Original value from request
        // Overide the value with hashed / encrypted value
        params.data.PASSWORD_FIELD = createHash('md5').update(password).digest('hex')
    }
}

You can read more about it here Understanding the different types/categories of Strapi Hooks