Can I display the colours from my default theme in the palettes?

Yes, you can add to or replace the palettes using the hooks brucebanner_default_bg_colors and brucebanner_default_text_colors.

For the background:

add_action( 'brucebanner_default_bg_colors', 'mytheme_brucebanner_default_bg_colors' );
function mytheme_brucebanner_default_bg_colors() {
	return [
			'#6d1e84',
			'#54366f',
			'#e40043',
			'#92BFC5',
			'#000000',
			'#ffffff',
		];
}

For the text:

add_action( 'brucebanner_default_text_colors', 'mytheme_brucebanner_default_text_colors' );
function mytheme_brucebanner_default_text_colors() {
	return [
			'#6d1e84',
			'#54366f',
			'#e40043',
			'#92BFC5',
			'#000000',
			'#ffffff',
		];
}