Banana,
Fatigue has already been cited as a factor in several incidents/accidents. In these cases, if fatigue didn't cause the accident, it was contributory in the events leading up to it. More worrying is your quote from management about wishing to employ more staff but citing economic reasons for not doing so. The Accountable Manager (at least under JAR 145) is entirely responsible for matters such as this. On their heads be it, theoretically at least, if something goes wrong as a result their parsimonious policies.