I know that with my gas range, whenever an igniter is used for any of the burners, it sparks them all. By guess, this is probably a safety feature in case any burner was turned on without igniting it. What I'm getting at though is that if you are able to ignite the oven by the burners, then it might be the same feature and in which case - there is gas in the oven that can be ignited. Gas should be either burning or cut off - the burner ignition should not affect it.
So, my guess is maybe a bad igniter in the oven? It might have a secondary igniter specifically for the oven that is not connected to the burners - this one could be bad or disconnected. So that when you're primarily lighting the oven, it might spark the burners (all connected as a primary igniter) or vise versa. Then the oven could have it's secondary igniter that is used for keeping the oven heated when it needs to reignite - without affecting the burners. Again, it's just a guess and you might find something about this in the manual or documentation for the range. You might also be able to see within the oven if there were a second igniter, however this might not be the case as it could be an internal wiring control that specifically ignites only the oven as well.
I don't think it'll be an easy fix, but it's most likely the igniter or rather the controls of it.