Don't think flat cap: think bonnet, instead, as in "Blue Bonnets over the Border". When anyone says flat cap, I (being part-Irish) think of a billed tweed cap. The cap in the painting is a most likely meant to be a tam or a bonnet, though it seems to be worn more in the style of a muffin cap. The painting looks a lot like period paintings and drawings that I've seen though instead of a bonnet the soldier is wearing a a cross between a feather bonnet and a bearskin shako. The look is pretty traditional, but not very widely worn as a style outside of re-enactor groups, so it might be a much more expensive look to recreate.