Looking at the first & second picture...
o Glass is fitted from the inside, retained by nailed on beading
o Corner measurement indicates a 6mm gap glass-to-beading
o Distant from that corner suggests the gap diminishes to 3mm

It does look like 4mm glass.

Glass rebate...
o Beading-to-front gap == 3mm for just below flush
o Beading-to-glass gap == 6mm
o Glass filled gap == 4mm
o Total rebate available for glass = 13mm

Could be higher - unknown is the glass-to-rebate re putty depth :-)

What is miss-fitted:
o Beading -- probably due to miss-cutting = forced-fit curvature
o Solution -- removal, shaving down & refitting

What is arguable:
o Glass thickness -- frankly 7-10mm SG would be better
o Solution -- available 13mm rules out DG, up to you re 7-10mm SG

I don't know the front glass-to-rebate putty thickness, I guess they
could have botched 4mm in there but if not then it's a bit too tight.

If the glazed area in the door is large, and it is subject to slam/wind, I
would put 7-10mm SG in. Small window in 4mm £6, 7mm £17, 10mm £21.
--
Dorothy Bradbury