Have you had a black powder, muzzle loading gunsmith take a look at your rifle?
As Rifleman1776 stated, you have a coned muzzle for ease in loading a patched ball. Determining the caliber of a rifle is best done by measuring as you have done. The ball size needed is determined by the diameter across the lands. I am assuming that you have an even number of lands and grooves. Let us know as this can make a slight difference in recommended ball size. Grimord's method is great for determining a bore diameter and counting the lands although I prefer a slightly longer just under bore diameter 11/32" or 5/16" brass so the removal is gentler on the barrel . You want a ball to shoot that is about 0.010" under the land to land diameter. Let's be gentle to your rifle so I would select a 0.360" diameter pure lead ball. I would use a thick patch of 0.022" denim or canvas as a patch material. The thicker patch material will fit into the grooves better and compress to the land to land diameter. To start out, any good oil such as olive oil, canola oil will do a good job for the trail shots. Powder of course should be 3fg black powder.
In the hand gun section there more than a couple of threads on calibers, especially 36. A 36 caliber revolver will require a 0.375" pure lead ball to fit the cylinder as no patching is required and lead is shaved off the circumference of the ball to seal the cylinder and hold the ball in place during firing of the other chambers in the cylinder. The land to land measurement in a revolver is the 0.360" and the excess lead will fill the grooves. A rifle or single shot 36 caliber pistol that requires a patched ball will also have a land to land diameter of o.360". However a patch is needed to seal the grooves and carry lubricant to the barrel while holding the ball and engaging the twist of the barrel to stabilize the ball. A ball 0.010" under the land to land diameter is a good choice for ease of loading and integrating with the patch for good performance.
Your rifle would be a 37 as it came from the gunsmith/builder. We can happily call it a 38. Fortunately we can find balls to fit your barrel. And as seldom as you are planning to shoot it, use a 0.360" ball.