This is a really neat piece, thank you for sharing it.
The first point to address is the "5 ADJ." marking. I have seen this marking on a number of watches (I have a 10.68Z) that were originally invoiced to a number of different countries (e.g. China, Canada, Germany). In my experience, these watches are not described as chronometers in Longines' archive so I do not consider them to be chronometers. Still, I find them intriguing as the archive does mention this "5 ADJ." marking in association with certain watches.
The second point to address is the cursive signature on the dial. Normally, Longines from the late 1930s do not have cursive signatures on their dials. However, there were exceptions (e.g. ref. 3582, other isolated examples). In my opinion, the dial of the watch in question looks original. The dial is uncommon, not only for its cursive signature, but for its combination of raised hour markers and black color.
A third point to address is the case-back markings. The markings found on the inside of the case-back are as expected. On the other hand, the cursive "Longines" marking on the outside of the case-back is uncommon. I have seen this marking on a few examples before.
A fourth point to address is the hands. To me, the gold-toned hour and minute hands appear appropriate for the period and dial design. The second hand is unusual both for its design (usually a circular counterpoise not oblong) and color (usually not red). Though unusual, red second hands are not unheard of on Longines from this period (e.g. a ref. 5193 invoiced to Egypt). Given the overall originality and condition of the watch, as well as the design and other anomalous attributes, I think that the red second hand could be original.
To tie everything together, below is a Longines with the same "5 ADJ." marking on its movement, a cursive signature on its dial, and a cursive "Longines" marking on its case-back. This watch was invoiced to Maier of China in June of 1939.
Regarding the price, while 650 GBP is not a bargain, I do not think that it is exorbitant either. The demand for 1930s and 1940s Longines is quite high, and examples in stainless steel with original black dials are not common.