-The Marines were not royal unit 1804(if you are a royal unit you have blue facing the marines didn't have blue facings till after 1804)
In the interest of nit-picking even further, I have to point out that the Marines were granted the 'Royal' title on 29 April, 1802. The order to adopt blue facings came through on 6 May, though it stated the new uniforms weren't to be adopted until 4 June to celebrate the King's birthday.
Another thing is that they're called the "1st Battalion" here, but British marine divisions simply had companies which weren't organised into battalions. No ship was allocated more than a company (they were allowed one marine per gun, plus an officer staff) so a battalion served no purpose. Sometimes temporary battalions were formed when several companies were serving on land together, but it wasn't a permanent establishment. There were three semi-permanent battalions formed in the Napoleonic era that wound up serving in the War of 1812, but they were quickly disbanded. In short, there may have been Marine battalions (there were two at Bunker Hill), but they was no "1st Battalion".