Ah Eddie - we are rare members of the brotherhood of the Cox Atmos! I had one fitted to my 1923 Invincible JAP (Australian-made 1000cc v-twin), and it ran "ok" without being fantastic over many thousands of miles. One afternoon I spent hours "tuning" the Cox Atmos, and eventually got a steady idle, fabulous response when the throttle was cracked open, and gobs of revs. On the stand. I headed off on a rally the next day and it wouldn't run down the road!! A friend, cleverer than me, had his JAP twin running beautifully on the same model carb. I've not seen a CA small enough for a 2 3/4 Douglas.
Ian the bike in your illustration is in the 1912 catalogue as the Model J, the predecessor to the 1913 Model P and the 1914 Model V. These models were listed by Douglas as "light touring", but in line with other manufacturers (e.g. Rudge, Triumph, Rover, Norton, ...) it could be (and clearly was) referred to (by others?) as a "TT Model", which was at the time an abbreviation for "our standard roadster but with no pedals, light mudguards and dropped handlebars". Remove the guards and tool box and they were often catalogued as the "Brooklands Racer".
This is where you should be careful. A "TT Model" pre-WW1 most likely used a bog-standard engine. To get go-faster goodies - a special cam, high compression piston etc. - your best bet would be to be a "special" customer of one of the big agents, of have friends at the factory. Is LB1120 really a "genuine TT Douglas"? Not in the strictest sense, because it doesn't have the very special parts that were seen on the "Works" bikes ridden by Bashall, Bailey et al.
But given that the Douglas catalogue pre WW1 didn't feature a "TT Model", LB1120 is the type of machine Douglas would supply to preferred customers that wanted to go racing. Pretty much a Model P with a special cam, perhaps special pistons, and certainly not a Douglas carburettor. The period photo in Eddie's thread shows an absolutely fabulous bell mouth on its non-Douglas carburettor.
To get a good idea of how far advanced the "Works" machines were in 1912, compare Bashall's 1912 TT bike to the bolt-on-manifold Model J.
Cheers
Leon