I think you could be a little less pre flop and still have the same result. If they are calling 3k they are calling 2.2k. And for sure you have to c-bet on the flop.
What I think you're missing is you need to know how this kind of calling station will act if she does have a weak A and hits it on the turn. Will she bet it if you check to her? Will she check the weak A? Will she raise if you bet? If you think she'll check it then you should check and keep the pot small. If she'll only call then make a small blocking bet and again, keep the pot smaller.
I hate inflating pots against calling stations because they so often can suck out. If you can get a free or cheap card, that's great, especially if she'll check it down twice here.
Don't try to represent an A here. It's the worst spot to do it. If she has nothing you're beating her anyways so why make her fold? If she has an A, she's never laying it down so you're just chip dumping. Betting like you have AQ only looses you money.