Child custody disputes are almost always reduced to liars contests, which Courts hate deciphering. The classic "he said, she said" is maddening to family law attorneys and judges because, without witnesses or documentation, the court does not know who to believe almost always resulting in the proverbial "A Pox On Both of Your Houses!"
These types of liars contests almost invariably consist of an accusation such as "he is always late picking up the children" or "I showed up to pick up the children and she stood me up." The best way to prove your fact is to do your exchanges at McDonalds.
When you go to McDonalds, you can order some coffee. When you pay, you get a receipt with the date and time on it demonstrating that you were there. Kids like McDonalds too. Never mind how unhealthy the food is. Many of these places have play areas and wifi. They are a lot less intimidating than the local police station too.
Many McDonalds restaurants are centrally located and are on main streets and therefore in safe areas of town. They are good places to exchange children. So you should propose that you and your ex do your pickups and drop offs at McDonalds. Order a cup of coffee. Save the receipt and bring it to court to demonstrate you were there on time to do the exchange.