According to Keith Over's Flags and Standards of the Napoleonic Wars, p 30. The Brunswick cavalry along with the Leib battalion and the Avant Guard did not carry colours.
The 10th Royal Veteran Battalion did have colours. A remnant of the center of the regimental colour is shown in a black and white photograph on page 184 of "A Scarlet Coat: Uniforms, Flags and Equipment of the British in the War of 1812" by Rene Chartrand, Service Publications, 2011. In the center of the regimental colour is a red circle outlined in a thin white border. Within the circle are four lines of text consisting of an X on top, ROYAL below, VETERAN below royal, and BATTALION on the bottom. The colour of the text is likely dark blue as per their facings. This circle is surrounded by the standard union wreath of roses, thistles and shamrocks common to most regimental colours. The main flag colour was dark blue as per the facings with a Union Jack in the upper left canton.
It is likely that the King's colour was in conformity with that of the line infantry with blue facings. So, the flag is probably a Union Jack with the regimental red circle in the center with surrounding wreath as per the regimental colour.
I can confirm that Stuart at Maverick Models is very helpful in creating published or unpublished flags. He has created many superb American militia flags for my Battle of New Orleans project.
Let us know your progress and good luck.