Kids will not be potty trained until they are developmentally ready. It's like a "door" inside of them. They won't be trained until the door opens. Your situation is not abnormal, so take heart, and try to stay emotionally "out of it" as best you can.
Many people have the idea that constipation during potty training is a control issue by the child. (this idea doesn't rule out that their truly can be a physiological problem, but they are rare). The more strong-willed the child, the more their need to be in control of their world, i.e., not pooping like mommy so frantically wants them to do. What they can control at this age is limited; refusing to poop is something they can.
Bring your child into the bathroom with you and let her see "how big girls" do it. This will appeal to her need to be "grown up". It also lets you start teaching her about good hygiene. (Fathers should do this with sons as well). Give a small reward for each success, like an M&M, but her real reward should be you pointing out her success at becoming more grown up.
Your daughter doesn't need laxatives and I'm sorry you received this advice. Talk to a dietician or research toddler diets on-line. There is tons of information, and you will find what fits her needs. I'm also an RN and know very well the down-side of laxative dependancy.
Staying out of it emotionally is vital. A reaction from the parent is bread and butter for the strong willed child. Love her patiently through this and be consistent. All of my children were not completely potty trained until 3 y/o, and I could not have cared less! I just never made it an issue. Don't let the super moms who have their kids trained by a year old (yeah, right!) get you down. The age is as varied as the kids are.
If you do stop the laxatives, be sure to do it with your pediatricians guidance, so you will be aware of any signs/symptoms your child may exhibit as her body adjusts itself to becoming laxative free.