LineRunner - but can you find a single example of a professional writer, eg on news media or in a novel, using "should of". There are usage sources that do provide a reliable guide.
I'm not an expert so I just see what's out there. I tried searching Google with the string "The National Lottery are" then replacing "are" with "is". The picture's mixed.
From the ITV website:
The National Lottery are holding an event in Telford today in the hope...
The National Lottery is looking for the eight remaining winners of...
From the Lottery Good Causes website:
With Andy Murray returning to Centre Court..., Roy's boys up against..., [other sporting examples]..., the National Lottery are behind them all the way.
Although the next sentence starts "The National Lottery has..." rather than "have"!
The National Lottery is today calling on people across the UK to vote for...
The Guardian's style guide says:
Nouns such as committee, family, government, jury, squad and team take a singular verb or pronoun when thought of as a single unit, but a plural verb or pronoun when thought of as a collection of individuals.
The BBC's style guide implies singular for organisations.
A lot of the "National Lottery is" examples are about the activity not the organisation, eg "The National Lottery is a great money-spinner". We're only interested in cases where it's the organisation: "The National Lottery is doing X" etc.
atticusclaw - you tell us that plural is not correct, but it would help if we knew what reasoning or evidence you are basing that on.