First off, you are ABSOLUTELY right that your child needs some kind of schedule. It only becomes more important as they become preschoolers -- because it helps give them structured areas of life that they can safely explore. The schedule, and discipline of a schedule, is reassuring and comforting to them.
So, I have two suggestions. The first thing is to try to remove the conflict between what you need and the best friend's opinion. A very simple approach is: "Your best friend's kids are not our kids."
That covers all differences in parenting styles. It covers the fact that his best friend is not the one who has to take care of YOUR child every day. It isn't saying that the friend is doing things right or wrong -- it's just saying that what works for his kids may or may not work for your kids.
My second suggestion is to use your pediatrician to your advantage. Make Daddy come to the next well (or sick) child visit, and ask the pediatrician about what to do about your child's behaviour at work when she has not gotten enough sleep, or how important a consistent schedule really is. Every pediatrician is going to tell you that your child needs sleep. They will probably tell you that a one year old should be sleeping about 12 - 13 hours a night, every night, and that interrupting that for anything but a major emergency is really hard on the child. Daddy's often will listen to a doctor before they will listen to Mommy. That's just how it works.
Finally, cultivating some new friends who have really well-behaved children and parent like you want to is a good thing. It gives you other couples for him to talk with who may be better influences in the long run :)