Sounds just like what we went through with our first baby. She would strain and strain and turn red and sometimes cry and it never amounted to much. It was terrible to see her going through that.
After getting a second opinion it was explained that babies really are not born KNOWING how to have a bowel movement and that sometimes it was necessary to teach babies/infants how to relax those muscles when having a BM. The doctor taught us how to help "stimulate" our daughter's rectum and she has been right as rain ever since. Of course, we had to proceed with caution (as would you) because true constipation can actually involve compacted stool and you wouldn't want the baby to be in pain after trying stimulation. We also had to give our daughter small drinks of water (per her doctor) to help soften her stool. A doctor can explain the stimulation in more detail to you as well as confirm whether or not your daughter's stomach is distended. Please check with the doctor before trying any of these things as I am not a doctor and am only explaining our situation.
While every child is different it is not even healthy for an adult to have a BM once every 10+ days. Not to second guess you or the Dr. or anything but I would get a second opinion.
Best of luck to you and your daughter. Hope you get through this quickly!