Whew, that was me EXACTLY. My little girl did the same thing, and would come and request a diaper so she could poop (I kept diapers for this reason). You are wise to give her the pull-up when she asks for one. My daughter would "hold out" for a diaper, and that did in fact lead to "bigger problems."
We tried every single thing we could think of. The only thing that worked was telling her that we would take her to get ice cream whenever she pooped in the potty. We told her this constantly, and finally, she asked to go poop in the potty. Then we took her for ice cream- at 8:30 at night! She went another week of asking for a diaper, then tried the potty again, and we again went for ice cream. It took a month or so, and we ate a LOT of ice cream, but it worked. I was worried that it would be hard to ween her off the ice cream reward, but she handled it fine. She would still say "Now we get to go get ice cream!" but she didn't notice that we never did anymore.
Good luck!