When I'm confronted with a complex problem, I favour simplification. Simplifying things aids focus and logic.
Four immediate simple points we can make, that I believe everybody (pro- or anti-Bamber, or like me, neutral) should agree on:
1. First, there is no direct forensic evidence linking Jeremy Bamber to the crime. This does not establish Bamber's innocence, nor does it infer guilt, but it is what it is and is worth re-stating.
2. Second, Jeremy Bamber's behaviour before and after the killings, including allegations that he went on an expensive holiday, are - even if true - of no relevance. They don't prove he killed anybody. Again, excluding this evidence is neutral as to the Crown and the defence.
3. Third, we can dismiss Julie Mugford's evidence entirely. Doing so is neutral as to the Crown and defence. It doesn't help Jeremy Bamber, nor does it hinder him. Neither does it help or hinder the Crown. Her evidence proves nothing one way or the other. At best, it's bad character evidence from which a rational jury might draw adverse inferences as part of understanding the big picture, but it's not probative and it can be neutralised by evidence of Julie Mugford's own bad character.
[Credit to David1819 for raising this] 4. As discussed below with David1819, it's accepted that Bamber could have entered and egressed the farmhouse via the windows, as Bamber admits that he could do so. Possibly at trial there was a ripe issue of how Bamber could have done so without leaving marks on the window frames, and perhaps more could have been made of this, but at this point it's not a very relevant issue. The appellate judges will simply recognise that Bamber could have entered and left via one of the windows and there may have been an unlocked window that could be opened from the outside without being forced, and Bamber could easily have 'secured' a window simply by closing or banging it shut in a way that would keep it shut. Nobody suggests he had to lock a window from the outside in order to fulfil his criminal scheme.
To amplify, especially with regard to point 3 above, let us consider an analogy, which I hope will help.
Many people think of cases like this one as akin to a house of cards. The thinking goes that if you remove one card, then the whole structure tumbles down. Thus, remove one piece of evidence, and the Crown's case collapses.
I would contend that in the case of Jeremy Bamber, the house of cards analogy is naive, if not wholly inaccurate. A better figuration would be to regard the Bamber case as a jigsaw puzzle, in which each node of incriminating evidence can be seen as a piece of the jigsaw. A playing card is structurally fungible - i.e. you can replace a card, and it won't matter if the new card is blue, black or pink, the house of cards remains in place as before, the essential structure intact. A jigsaw piece, on the other hand, is unique to that particular puzzle: a blank or 'grey' jigsaw piece is no good, ergo remove one piece of the jigsaw, and the puzzle fails.
Julie Mugford's evidence, in that figurative analogy, is the frame around the jigsaw. Remove the frame, and the puzzle remains in place, undisturbed. But remove a jigsaw piece, and the puzzle fails. It's the jigsaw pieces that need to be 'tested'.
So what are the jigsaw pieces?
The bases of Jeremy Bamford's conviction are circumstantial inferences that flow from the following elements [feel free to add to this list in the comments, if you are aware of any more):
(i). The forensic evidence found on the discovered sound moderator.
(ii). The witness evidence of the two individuals involved in discovering the sound moderator.
(iii). The lack of evidence (beyond Jeremy Bamber's own statements) of telephone calls, during the incident, from Nevill Bamber to either Jeremy himself or the authorities.
(iv). The view of the Crown's ballistics expert that Sheila's fatal injuries were not self-inflicted.
(v). The lack of any forensic evidence found on Sheila that would implicate her in a murder-suicide scheme.
(vi). It is apparent that Nevill had been involved in a physical struggle in the kitchen. [Note: I might move this to the 'Irrelevant' category, I will give it further thought].
[Let me know if you have additional points, and I will edit this and add them and gladly credit you].
Undermine just one of these, and the Crown's case fails. That's because, much like a jigsaw puzzle, each of these elements is non-fungible and interlocks with the others to form a cumulative 'puzzle', a picture of guilt.
The question we should ask and seek to answer is not, Is Jeremy innocent? or Is Jeremy guilty?, but rather, Is the conviction safe? We can only answer that by tackling (i) to (v) above (plus any additional points others can think of).
Tackling the incriminating evidence does not necessarily mean completely exposing it as false. All that's needed is reasonable doubt, which in the context of a criminal appeal means that the appellate judges will conclude that a rational hypothetical jury, on seeing the new evidence, could have come to a different verdict.
Again, note the use there of the word 'could' rather than 'would'. We are not here to decide what the jury at trial would or should have done if they had had the benefit of a leisurely retrospective, as we have. Instead, we must hypothesise whether particular new evidence could lead to a different verdict.