Wow. O.k. I haven't been on in a long time, and I don't normally post (or ever actually), but I think this is a fascinating topic. And I'll enter my humble 2 cents.
I think it's a moot point to argue why it happens, or if it should happen, it happens. It's bound to happen. It's happened before, it'll happen again.
I say, stop seeing her altogether and give her the heads up why (but probably not during your time together, that'd be weird). 'Cause if it's going to happen it'll happen, if not it wont. And by prolonging it, you're just going to end up, beating yourself up every time the hours up.
Then if it happens, cool; if not, well at least you know. And at the very least being open and honest about it shows respect and consideration for her. You leave the ball entirely in her court, with no pressure or awkwardness or hard feelings.
Most likely you'll get shot down, but hey; how different is that from the real world? :-P
But I think the crucial part is stop seeing her altogether. Because if you truly do care for her, and do want a relationship, Then continuing to see her as a client is a bit disrespectful to those feelings.