I don’t think the curve object itself has the options to do this directly.
Usually, people will use a different method to do what you are doing here. You could model a short section of the hedge, add an array modifier to repeat it and then use a curve modifier to make it follow the shape of the curve.
Yes, that’s what I was also thinking, but I also wanted to be a “big boy” and making everything procedurally, so I can change only one profile curve and everything follows.
I’ll try something else I currently have in mind, if that won’t work, I guess I’ll go for modeling those end caps manually like you pointed out.