His Wiki certainly reads like it's a real possibility he fabricated many, even most, of his 72 claimed victories, a total that would make him either the second or third allied ace of the War, depending on your count for Mick Mannock. But I don't have the background to interpret what I'm reading, here:
After years of controversy over Bishop's record, mainly because very few of his claimed victories were witnessed by anyone else or could be confirmed from the few surviving German records, the show led to an inquiry by the Canadian government in 1985. The Standing Senate Committee on Social Affairs, Science and Technology discredited the documentary, saying it was an unfair and inaccurate portrayal of Bishop. There is some dispute about whether Bishop or Mick Mannock had the highest score of any British Empire First World War fighter ace.[citation needed] The Canadian Encyclopedia states: "Investigation by a Senate sub-committee exposed a number of minor errors in this apparent 'documentary' and confirmed that statements had been wrongly attributed and incidents shifted in time for dramatic effect. However, the senators were unable to demonstrate that Bishop's claims were valid, and consequently recommended only that the film be labelled as docu-drama".[citation needed]
Some of Bishop's other claims have also been challenged. While combat reports and claims of both sides are littered with well-intentioned errors and accidental duplicate claims, there are two phases of Bishop's life in which German records can provide no supporting evidence. In his book on Victoria Cross airmen of the First World War, author Alex Revell quotes aviation historian Philip Markham's view about German records of the events of June 2, 1917 (the day of Bishop's VC award): "Not a shred of evidence to support Bishop's claims." Referring to Bishop's claims in early to mid-1918, Revell says another aviation historian, Ed Ferko, carried out extensive research on Germans records in 1987. Revell says that Ferko failed "to match a single victory claim made by Bishop against a known German loss for the day, time or place in question." However, distinguished First World War aviation historian Peter Kilduff says in his biography (Billy Bishop VC: Lone Wolf Hunter) that Bishop may have had as many as 21 matches in piecemeal German records. Kilduff, the author of 16 books, is renowned for his research in German records. Kilduff also makes a strong case for the unreliability of German records. He cites examples in which masses of data were destroyed by retreating German forces and instances of the German former air ministry having been guilty of "obfuscation" in denying losses when casualties had been incurred.[81]
What's the consensus? Was Billy Bishop a fraud?
For what it's worth, this note in the Wiki seems to suggest he was at least notorious amongst German pilots, a point against being a total fraud:
The successes of Bishop and his blue-nosed aircraft were noticed by the Germans, and they began referring to him as "Hell's Handmaiden". Ernst Udet called him "the greatest English scouting ace" and one Jasta had a bounty on his head.[43]
Having read a number of first hand accounts of WW1 combat it seems accurate kill counts were actually fairly hard to establish so I think some leeway in any aces kill count would be perfectly natural.
Open cockpit by Arthur Gould Lee has a section in it detailing how flights would try and add up kill counts. It was clearly a very messy business with many pilots having shot at the same target, shot at targets they believed may have been kills but having to break off due to being a target themselves, losing targets that appeared out of control in cloud cover, planes diving away which had far better rates of dive and they could not keep up with etc. Basically there were a lot of reasons for not being able to confirm kills and no doubt some kills counted as kills were not and visa versa.
Large dog fights were often complete chaos. The one certain thing they counted as a kill was to get a flamer. That's a plane going on fire at altitude.
I realise this does not answer the question specially on Bishop but I think it's important to realise these kill numbers had quite a bit of margin for error in them. Out right fraud would I think only really be possible on a lone mission as squadrons were debriefed on completion of a flight and had to discuss and agree on kill numbers. One thing in Arthur Gould Lee's account is it did seem flight leaders would often be given kills when they fired first and others also fired on the same target. This goes someway to explaining how some aces who were flight leaders racked up more kills than many.
Arthur Gould Lee also published their war time daily letters about their time in the RFC which are well worth a read. They are published under the title No parachute.
If you are all interested in the subject I would recommend reading both Gould's books. They are both freely available and they provide a very good level of detail on the day to day lives of a WW1 fighter pilot. Gould though only having 7 kills still saw a lot of action and had survived so many close calls it's amazing they survived the war. They had a lot of interesting observations to make on this subject.