If I read the OP correctly, you were already the company of choice for this client.
The PA likely had you on a list and called when service / upgrade was needed.
You would have been called by the client, if the PA never existed, so why should you be expected to give away 10%.
I would let her know you would be happy to just present the billing to her, and what she does with it (mark it up?) is up to her.
She said she does this with other contractors she brings in to work on her clients homes, implying she has other clients.
Maybe, if she can enlighten you on just how many other clients she has and what they could be worth to you, it might be worth working something out with her.

A previous company I worked for did something a little different with a couple of people that would bring us business. We used a point system. Every $10,000 worth of work generated X number of points. The points could eventually be redeemed for tech toys like a new TV, or some other electronic device they might have on their wish list.
One person did not really want anything new in tech, so the company paid for airfare to Europe, which is what they really wanted. I think the boss just used up some airmiles for that, so it did not really cost any out of pocket cash to keep someone happy.
There might be ways to deal with this, just might have to be creative.