As I understand it, the negative feedback loop provides a path for the output of the op amp to regulate the inverting pin. In the schematic U1G2 can do the same by controlling U2.
I think you're right about the meter, I tried the 5V regulator as xavier60 suggested and it messes with the regulation, guess I'll have yet another small transformer.
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David,
for the offset you mean something like the trimmer (R11) I had in the first schematic?