What makes me bit queasy about this case is that it's so easy for debators to go for stereotyping the mentally ill. It should not be forgotten that they are not more violent than the general population and that murderous behaviour is just as rare with them (and with the added factor of truly diminished responsibility). It's very easy to accuse and demonize Sheila. If she did this awful act, it really was an awful, sad tragedy.
This said I think the conviction is unsafe. Bamber might have done it (and he is not a very likeable character) but the police simply screwed up this case and there is plenty of reasonable doubt.
I don't think many here go for stereotyping the mentally ill. However there are certain features of the case that indicate a possible reconciliation between the actions of the killer and some of the facts. Your point is very interesting in that I recently read of the view of American Courts, Texas in particular, where mental stability is measured by times when a mentally unwell person is relatively stable, but not during an episode/crisis. If a person has periods of stability the Court will not support arguments against the death penalty in capital cases. That instability of course is sometimes the result of not taking meds or more likely an overpowering crisis, resulting in a mentally unwell person given a death sentence because at some point in the past having been able to be assessed as cognitively stable even if only for a short period.
Apart from other forensic factors that point to Sheila, her children in bed, the bible near by (on her chest) from memory, and on the floor next to her mother would not easily been contrived from a different killer.