Working with a contentious co-parent is a very challenging situation to be stuck in. You might want to continue the relationship because you feel it is best for the children, but you should think carefully about what is going on. There are several things that might come into the picture here.
One thing that you need to think about is what boundaries you have set. This factor alone has a big impact on how things are handled. If you have reason to believe that your ex is going to try to nitpick everything, make sure that you have a detailed and rigid parenting plan. This lays out what will happen and doesn't leave room for misinterpretation or negotiation.
Of course, there might be times when you will need to review the parenting arrangements and change them up some. For example, if an out-of-town family member is coming for a visit, the child may want to see that person. This might mean that you have to make a chance to the parenting time schedule. Hopefully, both parents can work as a team to make this happen instead of fighting against each other.
It might also help if you remember that every decision you make must be based on what is best for your children. No matter what your ex tries to do, don't let them rile you up. Instead, keep your cool and stay relaxed. This may help you to diffuse volatile situations. You should also be ready to take a step back when necessary. Taking the time to cool off and giving your ex the same opportunity can also be beneficial.