For those that have experience doing this, do you ask the girl to reach out to the booker on your behalf, or to let the booker know that you'll be reaching out to them?

I reach out to the girl and ask her if it’s OK if I use her as a reference with her booker. Then I tell the booker she’s expecting him to reach out to her.  

From very trusted mongers the procedure (obviously can slightly vary) is the following:
You talk to her and give her the name/number etc and ask her to validate you with PO. Then you text the PO and say that she was supposed to validate your number.  

helps to have their contact if they're popular (and indie) since they can book way in advance for you as well.

Early mornings are my favorite. I’ve had many a great appointment before the official start time because the girl knew me and cleared it with her booker.

And I have had varied levels of success with this. The problem with booking with the girl directly is you’re trying to make your appointment time fit in with the other bookings coming in from her booker, and while you may not give a shit if someone else’s appointment is being moved around, it still can complicate things with the booker, and by extension the org.  

if theyre from an independent org, i've swapped kakao and was told i can book far in advance since it's "her business". the time when a girl's hours are shortened randomly that week? regular clients

Personally, I wouldn't try this, at least in the bay. Imo it's a good way to get POs pissed at you. You're essentially bypassing org and PO fess.

I also oftentimes pass on a generous tip to the girls to provide to the org, back when this type of relationship was more common. I now keep to mainly 1 or 2 orgs in the Bay, and yes, I go through the POs which I am used to using. I will still rally with the girls for availability but will go through the POs to book