Went to specialist without referral, paid out of pocket, but (HMO) insurance FAQ (maybe) says I can go direct to specialist

I’m a little wary of contacting insurance directly first, I don’t want to accidentally trigger something that prevents coverage. Not sure whether I can say which insurance company?

A little background:

I’ve never seen a PCP as an adult

I scheduled an appointment with a doctor my sister strongly suggested, and confirmed they accept my insurance

On the day of the appointment, after filling out all of the paperwork, they informed me that they are a specialist and they wouldn’t be able to see me without a referral, unless I paid out of pocket

I agreed and paid out of pocket (for an office visit)

They scheduled a follow up appointment, but suggested I try to see PCP first to try and get a referral

While on my insurance website trying to find information about my PCP, I see a FAQ that I do not need a referral to see a specialist. Here is the exact wording:

Do I need a referral to see a specialist?

No. You can self-direct your care to a specialist of your choice. However, your benefits will be paid at a higher level when you get care from an in-network specialist.

Trying to dig through coverage documentation, though, I see this line:

All Covered Services (except in emergencies) must be provided by or through Your Participating Primary Care Physician/Practitioner, who may refer You for further treatment by Providers in the applicable network of Participating Specialists and Hospitals

My main question, is whether I can get reimbursed for (a covered portion of) the initial visit. And if so, what’s the best way to proceed? Do I try to get the doctor’s office to reverse the initial charge and bill insurance? Can I submit something to the insurance myself to try and get reimbursed?

Edit to add: I’m in Texas, if that matters