I've never had to ask a doctor to prescribe a generic EVER. If there is a generic it just gets substituted by the pharmacist because that's how things actually work. If there is a generic, you don't really have to do anything.
Whether or not the pharmacist is allowed or required to substitute a generic varies by state (e.g. it's mandatory in NJ and unlawful in OK, and up to the policy of the pharmacy in FL; and in states like HI/KY/NC/SC/TN the legality varies by drug).
It also varies from case to case (in this case, while both are epinephrine pens they aren't AB substitutable according to the FDA, so even states that allow some substitution for brand-name medications wouldn't allow it here)
There's literally no reason for a doctor to prescribe by brand name unless they believe it's one of the few cases where a particular brand is actually more effective: If the doctor prescribes the generic drug name, then it doesn't matter where you get it filled. You can get whatever brand is cheapest. If the doctor prescribes by brand name, then you're at the mercy of the local laws and regulations of the place that you fill it as to whether you can substitute or not.