Where to put code related to invariants

I am developing an small application, just to practice DDD. Afaik. invariants are the umbrella term of validation related to domain. So for example if I want to have only ucfirst names, then that is an invariant and I need validation in the property setter to make sure that this invariant is protected and make only the setter public, not the property itself.

My current problem is that I need to have usernames, which are generated from firstname, lastname, birthDate and probably some random stuff, if these are not unique enough. As far as I understand having a unique username is a domain concern, because it is very important if you want to login, or just bind some data to an actual user. So having a database table with pk on the username and wait for violating constraint error is just not enough, I should put some code into the domain, which makes sure that the generated username will be unique. I guess the code goes into the repository or into the entity somehow. Can you show me an example how this would look like in a real app? The domain and the app service implementation what I am curious of.

I might not be right about this. Every entity has a unique identifier and has repository, so probably I should put the username generation into the repository, and that's all. I might need to rephrase this for example what if we don't have a unique identifier, just a property which should be unique?

I made a little code draft:

class CustomerService {

    // ...

    public createCustomerFromNameAndYear(name, year){
        try {
            transaction.begin();
            username = generateUsername(name, year);
            while (repo.findByUsername(username))
                username = addSomeRandomChar(username, 1);
            password = "";
            password = addSomeRandomChar(password, 6);
            customer = new Customer(username, password, name, year);
            repo.save(customer);
            transaction.commit();
            dto = new CustomerDTO(customer.getUsername(), customer.getPassword());
            return dto;
        }
        catch(exception) {
            transaction.rollback();
            throw new ExceptionWithCauses(CUSTOMER_CREATION_FAILED, exception.getMessage());
        }
    }

}

If I put this into the app service, it will ensure that the name is unique unless if concurrence happens, but sending an error message is okay in that rare case. I am not sure whether this is okay. It does not feel right if the invariants must be parts of the domain code.

Another possible solution to have a create method in the repo.

class CustomerService {

    // ...

    public createUserFromNameAndYear(name, year){
        try {
            transaction.begin();
            customer = repo.create(name, year);
            transaction.commit();
            dto = new CustomerDTO(customer.getUsername(), customer.getPassword());
            return dto;
        }
        catch(exception) {
            transaction.rollback();
            throw new ExceptionWithCauses(CUSTOMER_CREATION_FAILED, exception.getMessage());
        }
    }

}

class CustomerRepository implements iCustomerRepository {

    // ...

    public create(name, year){
        username = generateUsername(name, year);
        while (this.findByUsername(username))
            username = addSomeRandomChar(username, 1);
        password = "";
        password = addSomeRandomChar(password, 6);
        customer = new Customer(username, password, name, year);
        this.save(customer);
        return customer;
    }
}

This might be better.