Just a note which may or may not be of interest. If you are speaking/writing to an American, you don't DO a course, you TAKE a course. In the same vein, you don't WRITE an exam, you TAKE an exam.
It appears that BrE is the only variant in which it's absolutely fine. You'll also hear "take a course" from some people here. However, a likely dialogue here would be:
Jane: I'm doing an Italian course this summer!
Fred: Great. Where are you doing it?
Jane: At the local college, every Tuesday evening.
Fred: I'd love to do a language course but I can't find any at a convenient time.
Jane: That's a shame.
Fred: It is. My work hours are changing next month so maybe I'll be able to find a course I can do after that.
I hope you are not saying BE is better than AE.
emsr2d2 here is a quote from your post
Then you use "do" in your dialog example.
That appears to imply to me you are suggesting this was something where American English that normally uses "take" instead of "do" cannot be used.
I showed that "take" works just fine.
I entirely accept (and said as much) that in the other variants "take" is preferable and I said that you will also hear "take a course" in the UK.