Thanks Flameport, your advice has been really helpful.
So I have temp connected the 2nd Bell Chime from the terminals within the 1st Bell Chime, and I can now get the 2nd Chime to quietly Dong (it doesn't Ding!).
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Am I right in saying that is due to the length of temp cable I am using (and a 2amp Transformer would fix that problem as you suggest?).
Of course, the above is superceeded by the fact I cannot get to the wiring that goes from the 1st Bell Chime to the Bell Push, as it is all buried under plaster and within the frame of my new Porch! Bugger!
Is there any way around this?