I do think that in reality IDDM will affect a career as a sturgeon. Particularly is it was poorly controlled. Lunch breaks can be erratic and there will come many a time I am sure that you will not be able to effectively control your diabetes due to more pressing immediate concerns.
If it can be assumed that failing to control your diabetes (in surgery) for prolonged periods is more likely to lead to consistent hyperglycaemic states; the long term affect of the diabetes is not going to lead to anything good. Forget for a moment the risk of going hypo because you haven't eaten enough or gone a bit happy go lucky on the insulin and consider a realistic 50-70 hour working week where there will be long period of time where you will not be able to stop and sort your diabetes out. >>> vascular disease, renal failure, neuropathy, retinopathy etc. Sure from an occupational health point of view it will be fine but the long term consequences of failing to control diabetes are severe.
As for a career in anaesthetics, that is achievable without too much concern. The periods where an anaesthetist absolutely cannot leave a patient tends to be quite short with longer periods being much less frequent. You spend much less time involved in sterile procedures as well, so can nip off for a minute in theatre or on the ICU/HDU.
I must also note that I am neither a doctor nor a medical student, so my post is purely speculative.