Hi there,
An oxypilot is a safety device !!!. It is designed to cut off EXACTLY as you describe.
As the air in the room is consumed over time, the flame on the oxypilot burner is designed to lift off and miss the thermocouple, thereby cutting off the fire.
It is very possible that the pilot is actually doing what it's supposed to do, Ie cut the fire off after 10 minutes of burning of the fire, as the oxygen levels have depleted.
The first thing you should be doing is to ensure there is adequate ventilation and that the flue is working properly.
It is a common misconception even amongst gas engineers that gas fires do not require ventilation, which they do. It is assumed that rooms have a certain ammount of adventitious ventilation which normally covers it, but in a tight room, then purpose provided ventilation may also be required.
If the ventilation and flue ARE is in order, you need to Check the flame to ensure it is heating the thermocouple properly..if it's only just touching, then the pilot needs a good clean out. Unfortunately modern oxypilots can't be taken to bits, so if it is choked at the injector, then a new pilot is indeed required.