I can't help with identifying if this is military issue or not but I can clear up a few points.
This is what is called a Balmoral style bonnet but does not have the standard rosette we use today.
Many Balmorals have some sort of dicing. This is the checkered pattern around the band. This dicing can be of many different colors including red/white, Red/black/white, etc.
This is not an uniform issue item for the police. It is also not the Sillitoe Tartan. Sillitoe is blue/white
The Scottish Tartans Authority has this to say about Sillitoe.
"This is the name given to the blue and white chequered band worn around their hats by many of the world's police forces. Strictly speaking it isn't a tartan and Sir Percy Sillitoe (Chief Constable of Glasgow) didn't design it - it had existed for about 100 years as an Heraldic symbol in many Scottish coats-of-arms. Highland soldiers are said to have woven white ribbons into their black hatbands, thus creating a chequered effect. Sir Percy - answering criticism that it was difficult for the public to differentiate between the police and bus conductors and other uniformed officials, introduced the three line chequered bands in 1932. The experiment was a success and the idea spread across the world with the ultimate accolade being its adoption by the European Union as the universal symbol of the police. Sir Percy was Director General of MI5 the UK's internal security service from 1946 to 1953."
I cannot comment on the usage of the word "Atholl" in reference to this type of Balmoral. I cannot find many contemporary photos of The Atholl Highlanders in anything but the fore and aft style bonnet called a Glangarry.
I did find photos of The Atholl Highlanders when they participated in the Braemar Gathering wearing Balmorals but without any dicing.
I can tell you that the current Atholl Highlanders are not part of the British Army. They are a ceremonial unit under the employ of Duke of Atholl and as such are the last European private army.
They would not get their uniforms from British Army stores.
It would seem possible that you have a bonnet from the Lovat Scouts. Many old photos show them in Balmorals similar to yours.
Caubeen_Lovat_Scouts_2.jpg
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