My cat Biscotti and I had this exact battle with a no-pets building about a year ago, so I can actually speak to this one.
What helped me understand it first was knowing that an ESA letter is a clinical document written by a licensed mental health professional, not just a certificate you buy online. So the legit services are the ones that actually have you go through a real evaluation with a therapist, not just a checkbox form.
I used RealESALetter.com, and the process was a same-day evaluation with a licensed therapist, then the letter came through pretty quickly after. My landlord accepted it without much pushback once I explained the Fair Housing Act basically requires them to consider reasonable accommodations for ESAs.
I wish you the best of luck with your apartment situation. It is more straightforward than it looks.