Can the hotel booking API price a room correctly when my customer is travelling with children?

Yes, a hotel booking API can price a room for a family, but only if it is told the children's ages before it searches.

When you run the search you send the number of adults, the number of children and the age each child will be at the time of the stay. The supplier uses those ages to decide whether a child can sleep free in the beds that are already in the room, whether a cot or an extra bed is needed and charged for, or whether the child counts as an adult for that room. That is why the same family can come back with quite different results at two hotels on the same street. If the ages are left out of the search, you will usually get an adults only rate that will not match what the guest is asked for at the desk.

Where it becomes less predictable is that each hotel sets its own rules and each supplier passes them on in its own way. Some hotels start treating a child as an adult at a younger age than others, some cap the number of people who may sleep in a room whatever their ages, and some accept the child but add a local charge on arrival that the API has no way to show you in advance. Family rooms and connecting rooms are also not always loaded for sale through the API even when the hotel has them, so your search may come back with fewer choices than the hotel's own website shows.

Before you quote a family, put the real ages into the search instead of guessing, read the room description and the rate conditions that come back with each result, and for a trip that matters or a party that is unusual, confirm the sleeping arrangement with the hotel in writing as well.