He was one of the few privateers to venture to Italy in the late 30s and circumstantial evidence from David Venables' "Racing 1500s" suggests that he might have been somewhat sympathetic to fascism: on the grid for the 1938 Gran Premio di Milano he was noted as having given an enthusiastic salute when introduced to the Propaganda Minister.
In November 1941, Motor Sport noted that "It is rumoured that Arthur Dobson is no longer in the R.A.F." I investigated this and found that he had been commissioned as a "hostilities only" Pilot Officer on probation on July 19th 1940, but that commission was "terminated on cessation of duty" on February 21st 1941. Whether this is perhaps connected to what Rob Young told us here I don't know, but at that time the RAF must have had a good reason to let him go: it could of course simply be some sort of injury, but in that case presumably he could have been transferred to other duties? The implication therefore seems to be that he was cashiered. (Pilot Officer does not necessarily imply flying duties, BTW, it's just the lowest commissioned rank in the RAF.)
A couple of further Motor Sport mentions show that "Arthur Dobson has to do with radio in a works of his own" (October 1944) and that, as a member of the pre-war committee of the BRDC, he was part of the caretaker committee formed in September 1945.
After that - nothing. R9B was reported as for sale in January 1942 and eventually went to Leslie Brooke (possibly via Reg Parnell?), but Arthur apparently just disappears until March 12th 1980. When he died in Battersea aged 65.

Such is the lack of information out there that his Wikipedia page has even been deleted on the grounds of "no assertion of notability", which - even though it was simply cribbed from Leif's Golden Era site - seems a trifle unfair

So, can anybody offer any more?