Leon,
There were other patents taken out by Barter, but they were not accepted/published.
By application number:
24859 06Dec01, J.J. Barter
20344 09Dec05, J.J. Barter
22642 06Nov06, J.J. Barter
1901 25Jan06, J.J. Barter & Light Motors
1902 25Jan06, J.J. Barter & Light Motors
I seem to remember in the Patent Office Journals (bound into annual volumes), there might have been the briefest of description of the patent application, but if so I did not make a note of it. These were all abandoned, probably due to an issue with prior art. The earliest one likely dealt with Barter's single cylinder engine design. Unfortunately when abandoned, all the submission material was discarded; a pity as for nothing else it would have given a view into what was on their mind at the time. However, being England, it would not surprise me if the documents are still laying lost in some repository at a forgotten branch office...
The model A is in the 1909 catalog, the earliest catalog I have a photocopy of. It is clearly the 'old' model with the very high diamond frame. The model B starting the trend to a lower frame and engine placement (albeit only slightly!) The description for the A states the engine is the same specification as the B (60x60), and naturally the B gets much more detail in the copy. But I am not sure that is the case as the exhaust port angles are different between the two illustrations. At first I thought this was just due to the 1909 A being an engraving and the 1909 B being a half-tone based on a photograph, but later on they show a section of an engine and a external photo, and again there are the two different port angles. Both still are screw in cylinders and the timing chest and engine's mounting to the frame look like they could be the same.
Though getting beyond the scope of this post, the 1910 catalog shows the model B continuing on, now as 'the old model' from the prior year, and the model C has pride of place as the new model. The model A has been omitted.
Unfortunately I have not seen and catalogs for the earlier Dougies (1907-08) or Light Motors. Just the odd tantalizing photo or article in the period magazine.
Back to Light Motors. Having a look again at the paperwork filed with the Trade Office shows the 1905 company was wound up in January 1907, and then immediately re-registered under the same name at a London address. The list of investors and their addresses indicates this is likely when Light Motors was sold to the "London interests". Total issued shares was now 10,000 pounds. J.J. Barter is still a share holder, but seems to have cashed-in a lot of his shares as he is down to owning 51 (vs. 800 in the incorporation of the 1905 company).
I also have a copy of the winding up document, and the 1907 company Light Motors Ltd was wound up in September of the same year.

Presumably this is when it was sold to Fairy Motors, but as I mentioned before, I did not come across any records for them. If they lasted until c1909, they must be
some paperwork for them surviving in the British Library, but I overlooked it. Unfortunately those records are not digitized, so I cannot check again from here on the other side of the Atlantic.
-Doug