The standard TSA policy within the USA, for Airport Security, is for TSA to honor requests for hand inspection tpo avoid CT scanners...no film thru hand baggage CT machines. Even international Security now generally recognizes the potential for damage to photographic films by the CT scanners placed at gate security in recent years, and will generally comply with request for hand inspection of film. 'Hand inspection' often entails the swabbing each item and then analysis of the swab for evidence of explosive chemical markers. As always, general policy may not be followed by every security agent, like someone not well trained or who is simply having a very bad day!...it might be necessary for intervention by a Supervisory security officer.
For the long-in-use X-ray machines, ISO 400 film is safe for passage in X-ray machines (but not CT), so disposable camera should withstand X-ray exposure without issue, even with multiple passess thru X-ray machines during the course of many legs of air travel.