On the fiddle
Engaged in a fraud.
'Fiddling' is usually meant to mean 'cheating in a petty way', perhaps falsifying one's expenses or not declaring all of one's taxable income. Of course, a fiddle is also a slang term for violin.
There are a couple of proposed derivations of the 'cheating' meaning of the phrase 'on the fiddle', each of them having supporters who are firm in their belief. Let's take the oldest first. The expression is said by some to derive from the Emperor Nero, who famously 'fiddled while Rome burned' and was a byword for corruption and dishonesty. The second suggestion is that the 'fiddle' was the name of the raised edge of the square wooden plates used by sailors. If a sailor took a normal amount of food he was said to have a 'square meal' and if his plate was overflowing he was said to be 'on the fiddle'.
As is often the case, I only set up those suggestions in order to knock them down. The Nero story is mere fancy. It may be a nice play on words that he was 'on the fiddle' in both senses, that is, he was both corrupt and a violinist (actually he wasn't even a violinist, there being no such instrument in Nero's lifetime, but let's not get sidetracked) but that's all this tale has going for it.
The culinary procedures on board sailing ships don't offer much of an explanation either. The idea that sailors' plates had raised edges and that these were called fiddles is quite incorrect. There were fiddles in sailing ships' galleys but those were arrangements of small posts and strings arranged around the edges of tables that were used to stop plates falling on the floor in rough weather.
If the above isn't enough to convince then the fact that 'on the fiddle' in the 'acting fraudulently' meaning is a mid-20th century idiom should clinch it. The expression wasn't known in the age of sail and certainly not in ancient Rome. A good place to look for a phrase like 'on the fiddle', with its association with minor crime, would be court records, and if the expression were in common use in English it might be expected to be found in the database of cases provided by the Central Criminal Court in England and Wales, commonly known as the Old Bailey. This is a comprehensive record of all the criminal cases brought to the court between 1674 and 1913, and no one was accused in The Bailey of being 'on the fiddle' during all that time.
The term 'fiddle' appears to have originated in America. It is recorded in an 1874 edition of John Hotten's Slang Dictionary:
Fiddle... In America, a swindle or an imposture.
Hotten also included this entry:
Fiddler... A sharper, a cheat; also one who dawdles over little matters, and neglects great ones.
'On the fiddle' was taken up by the British forces in WWII. It was well enough established in popular slang in the UK by 1961 for it to have been used as the title of a Sean Connery film and that is the first example of it that I've found in print. The plot involved a young Connery playing a streetwise rough diamond who runs various street scams while serving in the British army.